Items
Authored by W. Edwin Clark (a.k.a. W. E. Clark)
This page was taken from MathSciNet.
The other links on this page will not work. However if you or your
library has a subscription to MathSciNet you may go directly there
for this information.
CMP 1 798 943(2001:05) 05C69
(05C35)
Clark,
W. Edwin(1-SFL);
Suen,
Stephen(1-SFL);
Dunning,
Larry A.(1-BLGS-C)
Tight upper bounds for the domination numbers of graphs with given
order and minimum degree. II. (English. English summary)
Electron.
J. Combin. 7 (2000), no.
1, Research Paper 58, 19 pp. (electronic). ![[ORIGINAL ARTICLE]](to-original-article.gif)
{A review for this item is in process.}
2000m:0516805C69
Clark,
W. Edwin(1-SFL);
Suen,
Stephen(1-SFL)
An inequality related to Vizing's conjecture. (English. English
summary)
Electron.
J. Combin. 7 (2000), no.
1, Note 4, 3 pp. (electronic).
Summary: "Let $\gamma(G)$ denote the domination number
of a graph $G$ and let $G\square H$ denote the Cartesian product of graphs
$G$ and $H$. We prove that $\gamma(G)\gamma(H)\leq2\gamma(G\square H)$
for all simple graphs $G$ and $H$."
2000k:9405694B60
Clark,
W. Edwin(1-SFL);
Suen,
Stephen(1-SFL)
On the probability that a $t$-subset of a finite vector space contains
an $r$-subspace---with applications to short, light codewords in a BCH
code. (English. English summary)
Proceedings of the Thirtieth Southeastern International Conference
on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1999).
Congr.
Numer. 137
(1999),
139--159.
Summary: "Motivated by the problem of finding light
(i.e., low weight) and short (i.e., low degree) codewords in narrow-sense,
primitive BCH codes, we consider the problem of determining the probability
that a random $t$-set of vectors in an $n$-dimensional vector space over
${\rm GF}(q)$ contains an $r$-dimensional subspace (or affine subspace).
We find some bounds for this and similar probabilities and apply these
techniques to estimate how short a minimum weight codeword can be in a
narrow-sense BCH code."
\{For the entire collection see MR
2000k:05005.\}
2000a:2602326E99
(05C05)
Clark,
W. Edwin(1-SFL);
McColm,
Gregory L.(1-SFL);
Shekhtman,
Boris(1-SFL)
An application of spanning trees to $k$-point separating families
of functions. (English. English summary)
J.
London Math. Soc. (2) 58 (1998), no.
2, 297--310.
Let R be the set of all real numbers. Let ${\rm LS}\sb
k({R}\sp n,{\bf R})$ be the cardinality of a smallest $k$-point
separating family of linear functions from ${R}\sp n$ to R. For
smooth functions the cardinality is denoted by ${\rm DS}\sb k({R}\sp
n,{R})$. The authors establish that if $n,k\geq2$ then ${\rm LS}
\sb k({R}\sp n,{R})={\rm DS}\sb k({\bf R}\sp n,{R})=n(k-1)$.
The linear case result is also extended to a larger class of fields. Some
graph-theoretical results are invoked in the proofs.
Reviewed
by K.
Chandrasekhara Rao
99j:0510305C35
Clark,
W. Edwin(1-SFL);
Shekhtman,
Boris(1-SFL);
Suen,
Stephen(1-SFL);
Fisher,
David C.(1-COD)
Upper bounds for the domination number of a graph. (English. English
summary)
Proceedings of the Twenty-ninth Southeastern International Conference
on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1998).
Congr.
Numer. 132
(1998),
99--123.
Summary: "This paper presents several upper bounds
based on the degree sequence of a graph $G$ for the domination number $\gamma(G)$.
Our strongest result shows that the domination number of a graph $G$ with
$p$ vertices and minimum degree $\delta$ is at most $(1-S\sb \delta)p$
where $S\sb \delta=\prod\sb {k=1}\sp {\delta+1}(k/(k+1/\delta))$. This
supersedes bounds of Arnautov that $\gamma(G)\le(p/(\delta+1))H\sb {\delta+1}$
where $H\sb k=1+\frac12+\frac13+\cdots+\frac1k$. Slight improvements are
obtained in the case of regular graphs. Miscellaneous extremal results
are also obtained."
\{For the entire collection see MR
99i:00022.\}
99i:2008020M14
Clark,
W. E.(1-SFL);
Holland,
W. C.(1-BLGS);
Székely,
G. J.(1-BLGS)
Decompositions in discrete semigroups. (English. English summary)
Studia
Sci. Math. Hungar. 34 (1998), no.
1-3, 15--23.
Summary: "We prove that under some finiteness conditions
in a (not necessarily commutative and not necessarily cancellative) semigroup
every non-unit is a product of weakly irreducible elements. In commutative,
finitely generated semigroups, every infinitely divisible element is idempotent.
Without commutativity this is not true. An interesting open problem is
to find necessary and sufficient conditions for this implication."
Reviewed
by J.
L. Chrislock
Previous Review
Next Review
98m:0509605C35
Clark,
W. E.(1-SFL);
Shekhtman,
B.(1-SFL)
On the domination number of certain analogues of Kneser graphs.
(English. English summary)
Proceedings of the Twenty-eighth Southeastern International Conference
on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1997).
Congr.
Numer. 126
(1997),
175--181.
Summary: "Let $G$ be a multigraph whose vertices are
all $k$-dimensional subspaces of ${R}\sb n$. Two vertices $E\sb
1$ and $E\sb 2$ are adjacent iff $E\sb 1\cap E\sb 2\sp \bot=\{0\}$. We
show that the domination number $\gamma(k,n)$ of this graph is no more
than $k(n-k)+1$. This result complements the previous results in [W. E.
Clark and B. Shekhtman, Proc. Amer. Math. Soc. 125 (1997), no. 1, 251--254;
MR
97c:15003; Bull. Inst. Combin. Appl. 19 (1997), 83--92; MR
97g:05100], where similar estimates were proved for finite fields and
for an algebraically closed field. The proof relies on the methods of integral
geometry."
\{For the entire collection see MR
98h:05005.\}
Previous Review
Next Review
98m:0509505C35
Clark,
W. Edwin(1-SFL);
Dunning,
Larry A.(1-BLGS)
Tight upper bounds for the domination numbers of graphs with given
order and minimum degree. (English. English summary)
Electron.
J. Combin. 4 (1997), no.
1, Research Paper 26, 25 pp. (electronic).
The authors are interested in determining the maximum
possible domination number of a graph (not necessarily connected) with
a given number of vertices, say $n$, and with a specified minimum degree.
Using results of J. F. Fink et al. [Period. Math. Hungar. 16 (1985), no.
4, 287--293; MR
87e:05085], W. D. McCuaig and F. B. Shepherd [J. Graph Theory 13 (1989),
no. 6, 749--762; MR
90i:05053] and B. A. Reed [Combin. Probab. Comput. 5 (1996), no. 3,
277--295; MR
97k:05160] which deal with minimum degree one, two and three respectively,
they first determine the value exactly when the minimum degree is no more
than three. They also consider the case when the minimum degree equals
$n-k$ and $n$ is sufficiently large relative to $k$. For $n$ at most 14
(and for all but 6 values when $n$ is no more than 16) they give the exact
value as well.
Reviewed
by Bert
L. Hartnell
Previous Review
Next Review
97j:0503905C45
(05A10)
Clark,
W. Edwin(1-SFL);
Ismail,
Mourad E. H.(1-SFL)
Binomial and $Q$-binomial coefficient inequalities related to the
Hamiltonicity of the Kneser graphs and their $Q$-analogues. (English. English
summary)
J.
Combin. Theory Ser. A 76 (1996), no.
1, 83--98.
The Kneser graph $K(n,k)$ has as vertices all the
$k$-subsets of a fixed $n$-set and has as edges those pairs that are disjoint.
The reviewer and B.-L. Chen [J. Combin. Theory Ser. B 42 (1987), no. 3,
257--263; MR
89a:05100] proved that $K(n,k)$ is Hamiltonian if $\binom{n-1}{k-1}\leq
\binom{n-k}k$ when $n\geq 2k+1$ and $k\geq 1$. F. J. Zhang and X. F. Guo
[Sichuan Daxue Xuebao 26 (1989), Special Issue, 90--94; MR
91c:05123] proved that $K(n,k)$ is Hamiltonian connected if the above
inequality is strict. Chen and Lih defined $e(k)$ to be the minimum $n$
such that the inequality holds for $n\geq 2k+1$ and proved that $e(k)\leq
k(k+1)/2$. The authors of the paper under review obtain a more precise
approximation to $e(k)$. In addition they give an asymptotic formula for
the solution of $$k\Gamma(n)\Gamma(n-2k+1)=\Gamma\sp 2(n-k+1)$$ for $n\geq
2k+1$, as $k\to\infty$, when $n$ and $k$ are not restricted to take integer
values. The $q$-analogue of $K(n,k)$ is the graph $K\sb q(n,k)$ whose vertices
are the $k$-subspaces of the $n$-dimensional vector space $V(n,q)$ over
the finite field with $q$ elements and whose edges are those pairs having
the zero subspace as their intersection. The authors establish $q$-analogues
of the foregoing results of Chen-Lih and Zhang-Guo. They prove that for
$n\geq 2k$ and $k\geq 1$ the graph $K\sb q(n,k)$ is Hamiltonian if $\left[{n-1\atop
k-1}\right]\sb q\leq\left[{n-k\atop k}\right]\sb qq\sp {k\sp 2}$ and Hamiltonian
connected if the inequality is strict. They then show that the inequality
holds strictly if $n\geq 2k$, $k\geq 1$, and $q$ is any prime power. Hence
the graphs $K\sb q(n,k)$ are Hamiltonian and Hamiltonian connected.
Reviewed
by Ko-Wei
Lih
Previous Review
Next Review
97g:0510005C35
(05C70)
Clark,
W. Edwin(1-SFL);
Shekhtman,
Boris(1-SFL)
Domination numbers of $q$-analogues of Kneser graphs. (English.
English summary)
Bull.
Inst. Combin. Appl. 19
(1997),
83--92.
Summary: "We investigate the domination number and
total domination number of the graph $K\sb q(n,k)$ whose vertices are all
the $k$-subspaces of an $n$-dimensional vector space over a field with
$q$ elements and whose edges are the pairs $\{U,W\}$ of vertices such that
$U\cap W=\{0\}$. Bounds are obtained in general and exact results are obtained
for $n\geq k\sp 2+k-1$ and in other cases when $q$ is sufficiently large
relative to $n$ and $k$. We also consider analogous bipartite graphs."
Previous Review
Next Review
97c:1500315A03
Clark,
W. Edwin(1-SFL);
Shekhtman,
Boris(1-SFL)
Covering by complements of subspaces. II. (English. English summary)
Proc.
Amer. Math. Soc. 125 (1997), no.
1, 251--254.
Summary: "Let $V$ be an $n$-dimensional vector space
over an algebraically closed field $K$. Define $\gamma(k,n,K)$ to be the
least positive integer $t$ for which there exists a family $E\sb 1,E\sb
2,\cdots,E\sb t$ of $k$-dimensional subspaces of $V$ such that every $(n-k)$-dimensional
subspace $F$ of $V$ has at least one complement among the $E\sb i$. Using
algebraic geometry we prove that $\gamma(k,n,K)=k(n-k)+1$."
\{For Part I see the preceding review.\}
Reviewed
by B.
N. Moyls
Previous Review
Next Review
97c:1500215A03
Clark,
W. Edwin(1-SFL);
Shekhtman,
Boris(1-SFL)
Covering by complements of subspaces. (English. English summary)
Linear
and Multilinear Algebra 40 (1995), no.
1, 1--13.
Summary: "Let $V$ be an $n$-dimensional vector space
over a field $F$. We attempt to determine the least positive integer $\gamma=\gamma(k,n,F)$
for which there exists a family $U\sb 1,U\sb 2,\cdots,U\sb \gamma$ of $k$-dimensional
subspaces of $V$ such that for every $(n-k)$-dimensional subspace $W$ of
$V$ there is an $i\in\{1,2,\cdots,\gamma\}$ satisfying $U\sb i\oplus W=V$.
We find upper and lower bounds for $\gamma(k,n,F)$. In a few special cases
we find exact values."
Reviewed
by B.
N. Moyls
Previous Review
Next Review
96b:0512905C70
(05C35)
Chang,
Tony Yu(1-SFL);
Clark,
W. Edwin(1-SFL);
Hare,
Eleanor O.(1-CLEM-C)
Domination numbers of complete grid graphs. I. (English. English
summary)
Ars
Combin. 38
(1994),
97--111.
This paper concerns the domination numbers $\gamma\sb
{k,n}$ of the complete $k\times n$ grid graphs $(P\sb k\times P\sb n)$
for $1\le k\le 10$ and $n\ge 1$. These numbers were previously established
for $1\le k\le 6$. Here, dominating sets are presented for $5\le k\le 10$
and $n\ge 1$, giving new upper bounds for $\gamma\sb {k,n}$ for $7\le k\le
10$ and large $n$. Evidence which indicates that these bounds may be the
exact values of $\gamma\sb {k,n}$ is also discussed.
Reviewed
by Christine
M. Mynhardt
Previous Review
Next Review
96a:1101811B65
(05A19 05A30 11A07 33D20)
Clark,
W. Edwin(1-SFL)
$q$-analogue of a binomial coefficient congruence. (English. English
summary)
Internat.
J. Math. Math. Sci. 18 (1995), no.
1, 197--200.
This short and sweet paper is devoted to proving that
$[{na\atop nb}]\sb q\equiv [{a\atop b}]\sb {q\sp {n\sp 2}}\bmod \Phi\sb
n(q)\sp 2$, where $$\Big[{A\atop B}\Big]\sb q=\cases 0&{\rm if} B<0
{\rm or} B>A\\ \frac {(1-q\sp A)(1-q\sp {A-1})\cdots(1-q\sp {A-B+1})} {(1-q\sp
B)(1-q\sp {B-1}) \cdots(1-q)}&{\rm otherwise},\endcases$$ and $\Phi\sb
n(q)$ is the $n$th cyclotomic polynomial in the variable $q$.
The result is the $q$-analogue of the known binomial coefficient congruence
${pa\choose pb}\equiv{a\choose b}\bmod p\sp 2$ where $p$ is a prime.
The proof relies on a natural extension of the $q$-analogue of the Chu-Vandermonde
summation and the factorization of the Gaussian polynomial.
Reviewed
by George
E. Andrews
Previous Review
Next Review
95k:5200952A37
Clark,
W. Edwin(1-SFL)
Separating sets with parallel classes of hyperplanes. (English.
English summary)
Bull.
Inst. Combin. Appl. 13
(1995),
65--82.
Let $V$ be an $n$-dimensional vector space over a
field $F$ of order $q$. A hyperplane in $V$ is a translation of an $(n-1)$-dimensional
subspace $W$ of $V$ and the set of all translations of $W$ is called a
parallel class of hyperplanes. A parallel class $H$ is said to separate
a subset $S\subseteq V$ if each hyperplane in $H$ meets $S$ in at most
one point. A family $\scr P$ of parallel classes of hyperplanes is said
to be $k$-point separating if for every $k$-subset $S$ of $V$ there is
a parallel class in $\scr P$ that separates $S$. In the paper it is proved
that there is an integer $c(n,k)$ such that if $q\geq c(n,k)$ and $n,k\geq
2$ then any $k$-point separating family $\scr P$ contains at least $n(k-1)$
members. A partial result is also proved when $q<c(n,k)$.
Reviewed
by Bela
Uhrin
Previous Review
Next Review
94g:1500115A03
(05C99)
Clark,
W. Edwin(1-SFL)
Bounds on a class of partial partitions of a vector space over ${\rm
GF}(2)$: a graph theoretical approach. (English. English summary)
Linear
and Multilinear Algebra 32 (1992), no.
3-4, 225--235.
A collection $Q$ of linearly independent $w$-subsets
of the $n$-dimensional vector space $V(n)$ over ${\rm GF}(2)$ is called
a $w$-quilt if whenever $X$ and $Y$ are distinct elements of $Q$, then
$X$ is disjoint from the linear span of $Y$. The main results of the paper
are the following: (1) If $w\geq 6$, then every $w$-quilt $Q$ in $V(w+2)$
satisfies $\vert Q\vert \leq 21(2\sp {w-4}/w)+4$, where $\vert Q\vert $
denotes the cardinality of $Q$. (2) If $w\geq 12$, then every $w$-quilt
$Q$ in $V(w+2)$ satisfies $$\vert Q\vert \leq (21-\tfrac1{12}(1-(\tfrac14)\sp
{t-3})){2\sp {w-4}/w}+t+1,$$ where $t=[(w-\log\sb 2(w))/2]$.
Reviewed
by K.
Chandrasekhara Rao
Previous Review
Next Review
94b:0510305C35
Chang,
Tony Yu(1-SFL);
Clark,
W. Edwin(1-SFL)
The domination numbers of the $5\times n$ and $6\times n$ grid graphs.
(English. English summary)
J.
Graph Theory 17 (1993), no.
1, 81--107.
Let $G$ be a graph with vertex set $V$. A subset $D$
of $V$ is a dominating set if for every vertex $x$ not in $D$, there is
a vertex of $D$ adjacent to $x$. The order of the smallest dominating set
is called the domination number, and denoted $\gamma(G)$. Motivated by
the long-standing conjecture of Vizing that $\gamma(G\times H)\geq\gamma(G)\times\gamma(H)$,
numerous authors have determined $\gamma(G\times H)$ for various $G$ and
$H$. The reviewer and L. F. Kinch [Ars Combin. 18 (1984), 33--44; MR
87a:05087] determined $\gamma(P\sb m\times P\sb n)$ exactly, for $m=2,3$
and $4$ and all values of $n$. In this paper, the authors give an exact
formula for $\gamma(P\sb m\times P\sb n)$ with $m=5$ and $6$, and all values
of $n$. The formulas agree with those given by E. O. Hare ["Algorithms
for grid and grid-like graphs", Ph.D. Thesis, Dept. Comput. Sci., Clemson
Univ., Clemson, SC, 1989; per bibl.] for $n\leq 500$.
Reviewed
by Michael
Jacobson
Previous Review
Next Review
93h:1113911T99
(05B10 11B13)
Clark,
W. Edwin(1-SFL);
Pedersen,
John(1-SFL)
Sum-free sets in vector spaces over ${\rm GF}(2)$. (English. English
summary)
J.
Combin. Theory Ser. A 61 (1992), no.
2, 222--229.
A subset $S$ of an abelian group is said to be sum-free
if $a,b\in S$ implies $a+b\notin S$. Such a set is maximal sum-free (msf)
if it is not a proper subset of another sum-free set. Sum-free sets have
applications in areas such as Ramsey theory and coding.
If $V(n)$ denotes the vector space of dimension $n$ over ${\rm GF}(2)$,
it is shown that there are no msf sets $S$ in $V(n)$ if $5·2\sp
{n-4}<\vert S\vert <2\sp {n-1}$. For $n\geq 4$ there are msf sets
having cardinalities $5·2\sp {n-4}$ and $2\sp {n-s}+2\sp {s+t}-3·2\sp
t$ for $0\leq t\leq n-4$ and $2\leq S\leq [(n-t)/2]$. Also, if $S$ is an
msf set then $\vert S\vert \geq(\sqrt{2\sp {n+3}-7}-1)/2$. A computer search
has found all msf sets if $n\leq 6$ and numerous examples for $7\leq n\leq
10$.
Reviewed
by W.
A. Webb
Previous Review
Next Review
93g:9401294B05
(94B15)
Clark,
W. Edwin(1-SFL);
Dunning,
Larry A.(1-BLGS-C)
Partial partitions of vector spaces arising from the construction
of byte error control codes. (English. English summary)
Ars
Combin. 33
(1992),
161--177.
Let $H=\{H\sb 1,H\sb 2,\cdots,H\sb m\}$ be a partial
partition of the vector space $V$ defined over ${\rm GF}(q)$ with the $H\sb
i$ pairwise disjoint $w$-subsets and let $\langle H\sb i\rangle$ denote
the subspace of $V$ spanned by $H\sb i$. Of concern are partitions that
have one or more of the following properties: (i) Each $H\sb i$ is linearly
independent; (ii) $H\sb i\cap\langle H\sb j\rangle=\emptyset$ if $i\neq
j$; (iii) $\langle H\sb i\rangle\cap\langle H\sb j\rangle=\{0\}$ if $i\neq
j$; (iv) any two elements of $H\sb 1\cup H\sb 2\cup\cdots\cup H\sb m$ are
linearly independent; (v) any three elements of $H\sb 1\cup H\sb 2\cup\cdots\cup
H\sb m$ are linearly independent. This paper examines the following: (1)
the relationship between byte-error-control codes and partial partitions
with various subsets of the listed properties; (2) the largest partial
partitions, for fixed $r$ and $w$, having properties (i) and (iii); (3)
the largest partition (over GF(2)), called a quilt, for fixed $r$ and $w$,
having properties (i) and (ii). It also presents solutions to the following:
(4) Find the largest partial partition, for fixed $r$ and $w$, having property
(i) only; and (5) find the largest partition, for fixed $r$ and $w$, having
property (i), such that every component $H\sb i$ of $H$ is contained in
the set of columns of the parity-check matrix of a cyclic or an extended
cyclic code.
Reviewed
by J.
J. Stiffler
Previous Review
Next Review
93g:0511205C70
Clark,
W. Edwin(1-SFL)
Matching subspaces with complements in finite vector spaces. (English.
English summary)
Bull.
Inst. Combin. Appl. 6
(1992),
33--38.
Summary: "Let $C(n,q)$ denote the graph whose vertices
are the subspaces of the $n$-dimensional vector space $V$ over ${\rm GF}(q)$
and whose edges are the pairs $\{X,Y\}$ where $X\oplus Y=V$. We show that
$C(n,q)$ has a perfect matching if and only if $q$ is odd or $n$ is odd.
If $q$ and $n$ are even and $q>2$ we show there exist matchings which omit
only one vertex."
Previous Review
Next Review
92j:5101651E21
(05B25 51E22 94B05)
Clark,
W. Edwin(1-SFL)
Blocking sets in finite projective spaces and uneven binary codes.
Discrete
Math. 94 (1991), no.
1, 65--68.
Summary: "A 1-blocking set in the projective space
${\rm PG}(m,2)$, $m\geq 2$, is a set $B$ of points such that any $(m-1)$-flat
meets $B$ and no 1-flat is contained in $B$. A binary linear code is said
to be uneven if it contains at least one codeword of odd weight. If $B$
is a 1-blocking set in ${\rm PG}(r-1,2)$ and $\dim\langle B\rangle=r-1$,
any matrix $H$ whose columns are the vectors in $B$ is a parity check matrix
for an uneven binary code of length $n=\vert B\vert $, redundancy $r$,
and minimum distance at least 4; conversely, if $B$ is the set of columns
of the parity check matrix of such a code then it is a 1-blocking set.
Using this and results on uneven binary codes of minimum distance 4, we
show that there exists a 1-blocking set of cardinality $n$ if and only
if $5\leq n\leq 5·2\sp {m-3}$."
Previous Review
Next Review
91f:9402394B20
Clark,
W. Edwin(1-SFL);
Dial,
Gur(BR-FSC)
Remarks on the Sharma-Kaushik metrics for error-correcting codes.
J.
Combin. Inform. System Sci. 13 (1988), no.
3-4, 74--78.
Summary: "We show that the definition of a class of
metrics introduced by B. D. Sharma and M. L. Kaushik may be simplified
and we answer a question of Sharma concerning possible generalizations
of this class of metrics."
Previous Review
Next Review
91c:9402894B05
Clark,
W. E.(1-SFL);
Dunning,
L. A.(1-BLGS);
Rogers,
D. G.
Binary set functions and parity check matrices.
Discrete
Math. 80 (1990), no.
3, 249--265.
The authors study binary linear $(n,k,d)$ codes, where
$d$ is the minimum distance, $d\geq 4$ and $r=n-k$ is the redundancy, for
which one can construct a parity check matrix in which all of the columns
are of odd weight. This class of codes is put in correspondence with $(\delta,r)$-parity
standard systems, where $\delta=2\sp {r-1}-n$, $r\geq 4$.
The authors prove that, for $r\geq 4$, the value $n(r)=5\times 2\sp
{r-4}+1$ is a lower bound for the values $n$ such that every binary $(n,n-r,
d\geq 4)$ code has a parity check matrix formed only of odd-weight columns.
They also show that the value $\delta(r)=3\times 2\sp {r-4}-1$ is an upper
bound for the values $\delta$ such that every $(\delta,r)$-parity system
is standard.
This equivalence is applied to the study of binary codes with unequal
error protection, and new constructions and bounds are found for those
with minimum distance $d=3,4$.
Reviewed
by Josep
Rifa
Previous Review
Next Review
90f:9404994B40
(11T71)
Clark,
W. Edwin(1-SFL);
Lewis,
Larry W.(1-SFL)
Prime cyclic arithmetic codes and the distribution of power residues.
J.
Number Theory 32 (1989), no.
2, 220--225.
The authors discuss the weight distribution of prime
cyclic arithmetic codes, which are analogues of irreducible cyclic codes,
for which this problem has been studied in detail. The paper deals with
number theory, and the reader needs no background in coding theory.
Let $r$ and $n$ be integers $>1$, $m=r\sp n-1$, $p$ a prime divisor
of $m$, $a=m/p$, and $Z\sb m$ the ring of integers mod $m$. A prime cyclic
arithmetic code (with length $n$ and base $r$) is an ideal of $Z\sb m$
generated by $a$. Such a code can be taken equal to $Z\sb p$, and the following
norm (or weight) is defined in $Z\sb p$: let $\langle r\rangle$ denote
the subgroup of $U\sb p$, the group of nonzero elements in $Z\sb p$, generated
by $r$; $\Vert x\Vert $ is the number of elements of the coset $\langle
r\rangle x$ which lie in the interval $\scr M(p, r)=\{[p/(r+1)]+1$, $[p/(r+1)]+2,\cdots,\break
[rp/(r+1)]\}$.
The authors study the function $\Delta(p, r)=\max\{\vert \scr M(p, r)\vert
/d- \Vert x\Vert /x\in U\sb p\}$, where $d$ is the index of $\langle r\rangle$
in $U\sb p$, and they give upper bounds on $\Delta$, as well as conditions
on $r$ and $p$ to have $\Delta(p, r)=0$ (i.e., equidistant codes). Several
examples of equidistant codes, obtained by computer search, are presented.
Reviewed
by Antoine
Lobstein
Previous Review
Next Review
88h:9402794B15
Clark,
W. Edwin(1-SFL)
Cyclic codes over ${\rm GF}(q)$ with simple orbit structure.
Discrete
Math. 61 (1986), no.
2-3, 151--164.
Let $C$ denote a cyclic $(n,k)$ code over $F={\rm
GF}(q)$ having parity check polynomial $h(x)$ of degree $k$. The author
assumes that $n$ is the multiplicative order of $x$ modulo $h(x)$; by $R=F[x]/(h(x))$
he denotes the parity check algebra of $C$. If $R\sp *$ is the unit group
of $R$ and $\langle x\rangle$ is the subgroup of $R\sp *$ generated by
$x+(h(x))$, he calls $h(x)$ a simple orbit structure if $R\sp *=F\sp *\langle
x\rangle$. He calls $h(x)$ local if $R$ is a local ring. The author gives
a description of when $h(x)$ is local and has simple orbit structure in
terms of $q$ and the polynomial $h(x)$. In addition, he shows how to compute
the weight classes for a cyclic code having simple orbit structure.
Reviewed
by Eugene
Spiegel
Previous Review
Next Review
87c:9405994B40
Clark,
W. Edwin(1-SFL);
Liang,
Joseph J.(1-SFL)
Equidistant binary arithmetic codes.
IEEE
Trans. Inform. Theory 32 (1986), no.
1, 106--108.
Author summary: "Let $C(B)$ denote the binary cyclic
$AN$ code with generator $A$, where $AB=2\sp n-1$. It is known that $C(B)$
is equidistant if $B$ is a prime power $p\sp k$, where either 2 or $-2$
is primitive modulo $B$ provided $p\equiv1 ({\rm mod}\,3)$ if $k>1$. It
is conjectured that these are the only $B$ such that $C(B)$ is equidistant.
We have verified this for $B<100\,000$. We establish several results
that further limit the possibilities for counterexamples to the conjecture."
Previous Review
Next Review
82k:3900339B05
Clark,
W. E.; Mukherjea,
A.
Comments on a functional equation.
Real
Anal. Exchange 6 (1980/81), no. 2, 192--199.
The authors study all solutions of (1) $f(x+s)=af(x)$,
$f(x+t)=bf(x)$, where $a,b,s,t$ are real numbers, $a>0$, $b>0$, and $s/t$
is irrational. They note that they were motivated by wanting to know conditions
on the equation $f(x+1)=2f(x)$ to ensure that any solution equals $2\sp
x$ a.e. They also observe that (1) with $a=1=b$ appears in a problem of
W. Rudin [ Real and complex analysis, second edition, McGraw-Hill, New
York, 1974; MR
49
#8783]. Their results: (a) If there is a solution of (1) which is positive
at some point and bounded above on some interval, then $a\sp t=b\sp s$;
(b) If $a\sp t=b\sp s$, then every measurable solution of (1) equals $ca\sp
{x/s}$ a.e.
They also exhibit a solution of (1) which is not continuous a.e., but
they are unable to determine whether their example is measurable. They
assert that they have some generalizations to metric topological groups,
and they provide one such result.
As would be expected, a Hamel basis for the reals figures prominently
in their arguments.
Reviewed by R. A. Rosenbaum
Previous Review
Next Review
81k:9403794B40
Clark,
W. Edwin; Liang,
J. J.
On block irreducible forms over Euclidean domains.
Internat.
J. Math. Math. Sci. 3 (1980), no. 1, 15--28.
For elements of a Euclidean ring $R$ with a real valuation
$v(R)$, the authors establish a new canonical form, which generalizes previously
known forms, that is valid for integers with an arbitrary radix $r$ and
for Gaussian integers with radix $r=±1±i$. The authors call
this more general canonical form with radix $r$ of $R$ a block irreducible
form. They prove that this form is unique and minimal and thus allows us
to determine the arithmetical weight of an element with radix $r$. They
also prove a theorem on the existence in $R$ of block irreducible forms
and give a constructive algorithm for their determination.
Reviewed by V. A. Arakelov
Previous Review
Next Review
56 #1150894A10
Clark,
W. Edwin; Liang,
J. J.
Weak radix representation and cyclic codes over Euclidean domains.
Comm.
Algebra 4 (1976), no. 11, 999--1028.
Authors' summary: "We show that, to some extent, the
two theories cyclic arithmetic coding and cyclic polynomial coding can
be subsumed under a `unified' theory based on a more or less arbitrary
Euclidean domain. In particular, we show that Hamming weight (for cyclic
polynomial codes) and arithmetic weight (for cyclic arithmetic codes) are
special cases of a more general notion of weight defined for Euclidean
rings. Of fundamental importance in this development is a generalization
of the notion of radix representation for integers which we call a weak
radix representation. For several specific Euclidean rings we prove the
existence of certain unique canonical weak radix representations for all
ring elements. We show, for example, that every Gaussian integer has a
unique representation of the form (1) $a\sb nr\sp n+\cdots+a\sb 1r+a\sb
0$ where $r=i-1$, each $a\sb j$ is zero or a unit, and whenever $a\sb j\neq
0$, then $a\sb {j+1}=a\sb {j+2}=0$. This is analogous to a known result
of Reitwiesner that each rational integer has a unique representation of
the form (1) where $r=2$, each $a\sb j$ is $0$ or $±1$, and whenever
$a\sb j\neq 0$, $a\sb {j+1}=0$."
Previous Review
Next Review
56 #263394A10
Clark,
W. Edwin
Equidistant cyclic codes over ${\rm GF}(q)$.
Discrete
Math. 17 (1977), no. 2, 139--141.
Author's summary: "It is proved that a cyclic $(n,k)$
code over $\text{GF}(q)$ is equidistant if and only if its parity check
polynomial is irreducible and has exponent $e=(q\sp k-1)/a$ where $a$ divides
$q-1$ and $(a,k)=1$. The length $n$ may be any multiple of $e$. The proof
of this theorem also shows that if a cyclic $(n,k)$ code over $\text{GF}(q)$
is not a repetition of a shorter code and the average weight of its nonzero
code words is integral, then its parity check polynomial is irreducible
over $\text{GF}(q)$ with exponent $n=(q\sp k-1)/a$ where $a$ divides $q-1$."
Previous Review
Next Review
54 #276418H40
(20M25 55J10)
Clark,
W. Edwin
Cohomology of semigroups via topology--with an application to semigroup
algebras.
Comm.
Algebra 4 (1976), no. 10, 979--997.
Let $S$ be a semigroup with 0. Let $B(S)$ denote the
chain complex whose $n$th component is the free abelian group generated
by the set of all $n$-tuples $(s\sb 1,\cdots,s\sb n)$, where $s\sb i\in
S$ and $s\sb 1s\sb 2\cdots s\sb n\neq 0$. The boundary operator is defined
as it is for the group case. Although the usual cohomology groups of a
semigroup with zero are trivial, that need not be the case for the various
homology and cohomology groups derived from this complex.
For a field $F$ let $F\sp *$ denote the multiplicative group of nonzero
elements. Each cocycle $z\in Z\sp 2(S,F\sp *)$ gives rise to an $F$-algebra
$zF[S]$ with basis $S-\{0\}$ and multiplication given by $s*t\equiv z(s,t)st$
if $st\neq 0$ and 0 otherwise. This algebra is referred to as the twisted
semigroup algebra of $S$ over $F$ determined by the cocycle $z$. If two
cocycles $z$ and $w$ differ by a coboundary then $zK[S]$ and $wK[S]$ are
functorially isomorphic. Other relationships between cohomology groups
of $S$ and twisted semigroup algebras are derived in the paper.
By a matrix units semigroup $S$ is meant a subsemigroup of $\{e\sb {ij}\colon
1\leq i,j\leq n\}\cup\{0\}$ which contains 0 and all $e\sb i=e\sb {ii}$
(where $e\sb {ip}e\sb {pj}=e\sb {ij}$ and $e\sb {ip}e\sb {qj}=0$ for $p\neq
q$). Such semigroups are in one-to-one correspondence with the quasi-orderings
on the set $\{1,\cdots,n\}$. The order is defined by $i\leq j$ if $e\sb
{ij}\in S$. Two "oriented simplicial complexes" are obtained from this
quasi-ordered set which have the same cohomology groups as $S$ for $n\geq
2$. Conversely, every simplicial complex can be associated with some matrix
units semigroup which has the same cohomology. The paper closes with some
examples and remarks.
Reviewed by Jimmie D. Lawson
Previous Review
Next Review
52 #1685094A10
Clark,
W. Edwin; Liang,
J. J.
On arithmetic weight for a general radix representation of integers.
IEEE
Trans. Information Theory IT-19 (1973), 823--826.
The expression $N=\sum\sb ib\sb ir\sp i$ is said to
be a modified radix-$r$ form for the positive integer $N$, where $r$ is
an arbitrary integer greater than 1, if the coefficients $b\sb i$ are integers
satisfying the condition $-r<b\sb i<r$. It is said to be a generalized
nonadjacent form (GNAF) if the coefficients $b\sb i$ satisfy the following
two conditions: (1) $\vert b\sb i+b\sb {i+1}\vert <r$ for all $i$, and
(2) $\vert b\sb i\vert <\vert b\sb {i+1}\vert $, if $b\sb ib\sb {i+1}<0$,
where $\vert b\vert $ denotes the absolute value of $b$. A GNAF is proved
to be unique, and the arithmetic weight of an integer is shown to be equal
to the number of nonzero terms in the form. Two algorithms are presented
for the computation of this form. If $r=2$, the GNAF coincides with the
well-known modified binary nonadjacent form. The results are of interest
in the theory of arithmetic codes.
Reviewed by G. M. Tenengolc
Previous Review
Next Review
52 #584722A15
Clark,
W. E.; Mukherjea,
A.; Tserpes,
N. A.
Is topologically simple simple?
Semigroup
Forum 11 (1975), no. 1, 90--93.
A topological semigroup is simple if it contains no
proper ideal and topologically simple if it contains no proper closed ideal.
Motivated by a conjecture of H. L. Chow [Amer. Math. Monthly 82 (1975),
155--156; Zbl 302 \#22007] that a topologically simple subsemigroup of
a compact semigroup must be simple, the authors first give a counterexample
to the conjecture but show that it holds for locally compact subsemigroups.
This follows from the more general result that, in a locally compact completely
simple semigroup with compact maximal groups, each locally compact subsemigroup
is completely simple.
Reviewed by K. N. Sigmon
Previous Review
Next Review
50 #1243394A10
Clark,
W. Edwin; Liang,
J. J.
On modular weight and cyclic nonadjacent forms for arithmetic codes.
IEEE
Trans. Information Theory IT-20 (1974), 767--770.
The modular weight of an integer $a$, $0\leq a<m$,
used previously by T. R. N. Rao and O. N. Garcia [same Trans. IT-17 (1971),
85--91], was defined to be $w\sb m{}\sp *(a)=\min\{w(a),w(m-a)\}$, where
$w(·)$ is the arithmetic weight with radix $r$, $r$ an integer not
less than 2. In the present paper this is modified to $w\sb m(a)=\min\{w(x);x\equiv
a \text{mod}\,m\}$ and it is shown that for $m=r\sp n$, $r\sp n-1$ or $r\sp
n+1$, $w\sb m{}\sp *(a)=w\sb m(a)$, while if $m=r\sp n-1$ or $r\sp n+1$
then $w\sb m(ra)=w\sb m(a)$. For some values of $m$, $w\sb m{}\sp *$ may
fail to be a metric, a defect which $w\sb m$ does not have. The notion
of a modular cyclic generalized nonadjacent form for $m=r\sp n-1$ is introduced.
An algorithm is given for computing the modular cyclic GNAF for any integer
$N<r\sp n-1$ from its radix $r$ representation. The number of nonzero
digits in the modular cyclic GNAF of an integer $N$, $0\leq N<m$, $m=r\sp
n-1$ is shown to be its modular weight. The use of residues in computing
modular weight for $m=r\sp n-1$ is discussed.
Reviewed by Ian Blake
Previous Review
Next Review
49 #267912C05
(13B25)
Clark,
W. Edwin; Liang,
Joseph J.
Enumeration of finite commutative chain rings.
J.
Algebra 27 (1973), 445--453.
A commutative chain ring is a commutative ring with
identity, ideals of which are linearly ordered by inclusion. For example,
${Z}\sb q[x]/(f(x))$ is a finite commutative chain ring, where $q=p\sp
n$ is a prime power, $f(x)$ is monic of degree $r$ (say) and irreducible
modulo $p$. This ring, which appears in an early work of W. Krull [Math.
Ann. 92 (1924), 183--213, Jbuch 50, 72], is uniquely determined by $p$,
$n$ and $r$, and is denoted by $\text{GR}(p\sp n,r)$. Every finite commutative
chain ring $R$ is easily seen to be of the form $S[\theta]$, where $S$
is a coefficient ring and $\theta$ is a generator of the unique maximal
(principal) ideal $N$ of $R$. It is shown that $S\cong\text{GR}(p\sp n,r)$,
$\theta$ satisfies an Eisenstein polynomial equation $\theta\sp k=p(\sum\sb
{i=0}\sp {k-1}a\sb i\theta\sp i)$ and $p\sp {n-1}\theta\sp t=0$, where
$p\sp n$ is the characteristic of $R$, $p\sp r$ is the order of $R/N$,
$k$ is the largest integer ($\leq$ the index of nilpotency $m$ of $N$)
such that $p\in N\sp k$ and $t=m-(n-1)k$. The authors' main aim is to find
the number of isomorphism classes of rings $R$ with given invariants $p,n,r,k$
and $t$. When $n=1$, there is one such class $R=S[\theta]$ with $S\cong\text{GF}(p\sp
r) (=\text{GR}(p,r))$ and $\theta\sp k=0$.
Assume that $n>1$. $R$ is called a pure chain ring when $\theta$ can
be so chosen that the polynomial equation is of the form $\theta\sp k=pu$
with $u$ a unit of $S$. The main result is that there are at least $\sum\sb
{c\vert d}\phi(c)/\tau(c)$ isomorphism classes of pure chain rings, where
$d=(k,p\sp r-1)$, $\phi$ is the Euler $\phi$-function and $\tau(c)$ is
the order of $p$ in the group of units of the ring ${Z}\sb c$. Moreover,
if $(p,k)=1$ then all chain rings are pure and this lower bound is attained.
A theorem of Krull [op. cit.] for the case in which $R/N$ is algebraically
closed is extended to give necessary and sufficient conditions for one
isomorphism class corresponding to given invariants in the general case.
Reviewed by D. Kirby
Previous Review
Next Review
48 #1120016A44
Clark,
W. Edwin; Drake,
David A.
Finite chain rings.
Abh.
Math. Sem. Univ. Hamburg 39 (1973), 147--153.
A ring with 1 is called a chain ring if its lattice
of left ideals forms a chain, and it is known that a finite chain ring
is a local uniserial ring and the lattice of its right ideals forms a chain,
as well. L. A. Skornjakov [In memoriam: N. G. Cebotarev (Russian), pp.
75--88, Izdat. Kazan. Univ., Kazan, 1964; MR
34
#190] showed that a finite chain ring is a homomorphic image of a ring
of the form $\langle x,y\rangle/I$, where $\langle x,y\rangle$ is the ring
of polynomials in the non-commuting indeterminates $x,y$ over $Z\sb {p\sp
n}$ and $I$ is an ideal generated by four elements of specified type. In
the paper under review, the authors give a more revealing description of
finite chain rings: Let $R$ be a finite chain ring of characteristic $p\sp
n$. Then $R$ contains a unique (up to isomorphism) coefficient ring $S$
which is isomorphic to a Galois ring $\text{GR}(p\sp n,r)$ [cf. the first
author, Proc. Amer. Math. Soc. 33 (1972), 25--28; MR
45
#3481]. Let $m$ be the nilpotency index of the radical of $R$. Then
there exists an element $a\in R$, and positive integers $t\leq k$ with
$m=(n-1)k+t$, such that (1) ${}\sb SR=S\oplus Sa\oplus\cdots\oplus Sa\sp
{k-1}$; (2) $a\sp k=pu$ with a unit $u\in R$; (3) $Sa\sp i\cong S (1\leq
i\leq t-1)$ and $Sa\sp i\cong Sp (t\leq i\leq k-1)$. Moreover, specializing
to the commutative case, they show that a finite commutative ring is a
chain ring if and only if it is a homomorphic image of an Eisenstein extension
of a Galois ring.
Reviewed by H. Tominaga
Previous Review
Next Review
47 #21013D15
Clark,
W. Edwin; Bergman,
George M.
The automorphism class group of the category of rings.
J.
Algebra 24 (1973), 80--99.
Ausgangspunkt dieser Arbeit bildet die kategorietheoretische
Charakterisierung von Ringeigenschaften. Dies ist nicht immer moglich,
aber die Invarianz von solchen Eigenschaften bezuglich der Ringautomorphismen
gewahrleistet dies. Daher wird die Struktur der Automorphismen-klassengruppe
fur gewisse Kategorien von assoziativen $R$-Algebren bestimmt, wobei $R$
ein kommutativer Integritatsbereich mit Einselement ist. Die Idee fur die
Strukturtheorie in diesen Kategorien ${A}$ besteht im folgenden:
Sei $\langle x\rangle$ die freie Algebra mit einem erzeugenden Element
in ${\bf A}$. Fur jedes Objekt $A\in{A}$ betrachtet man die Menge
der Homomorphismen $[\langle x\rangle,A]$. Mit Hilfe einer Koalgebrastruktur
von $\langle x\rangle$ in ${A}$ erhalt man eine Algebrastruktur
fur die Mengen $[\langle x\rangle,A]$. Dadurch bekommt man einen Funktor
${\bf A}\rightarrow{A}$, der aquivalent ist mit dem identischen
Funktor. Eine Reihe von interessanten Beispielen, Gegenbeispielen und Problemen
beendet die Arbeit.
Reviewed by W. Vogel
Previous Review
Next Review
45 #348116A44
Clark,
W. Edwin
A coefficient ring for finite non-commutative rings.
Proc.
Amer. Math. Soc. 33 (1972), 25--28.
Following G. J. Janusz [Trans. Amer. Math. Soc. 122
(1966), 461--479;
MR
35
#1585] we let $\text{GR}(p\sp r,r)=Z\sb {p\sp n}[x]/(f(x))$ where $f(x)$
is monic and irreducible modulo $p$, and call such a ring a Galois ring
of characteristic $p\sp n$ and $\text{rank}\,r$, which is determined up
to isomorphism by $p,n$ and $r$. The principal theorem of the paper under
review is the following: A finite $p$-ring (a finite ring whose additive
group is a $p$-group) contains a unique (up to isomorphism) subring such
that $S/pS\cong R/\text{rad}\,R$. Moreover, $S$ is a direct sum of full
matrix rings over Galois rings. This is a generalization of the previous
results for finite commutative $p$-rings and for finite $p$-rings of characteristic
$p$. The proof of the theorem is reduced to the case of a finite local
$p$-ring.
Reviewed by H. Tominaga
Previous Review
Next Review
44 #686720.92
Brooks,
Burrow P., Jr.; Clark,
W. Edwin
On the categoricity of semigroup-theoretical properties.
Semigroup
Forum 3 1971/1972 no. 3, 259--266.
Die Autoren untersuchen die Frage, wann isomorphie-invariante
Eigenschaften von Halbgruppen in kategorischer Sprache beschrieben werden
konnen. Dazu wird (wie z.B. auch im Falle von Gruppen) die freie monogene
Halbgruppe $\langle x\rangle$ kategoriell charakterisiert, und zwar als
ein spezielles projektives Objekt. Durch Einfuhrung einer sog. Co-Halbgruppen-Struktur
kann nun die Menge der Morphismen von $\langle x\rangle$ in eine Halbgruppe
$A$ selbst zu einer Halbgruppe gemacht werden. Diese Halbgruppe erweist
sich stets als isomorph oder stets als anti-isomorph zu $A$. Hieraus folgt,
dass die obige Beschreibung immer moglich ist, wobei aber einseitige Eigenschaften
(z.B. die Existenz eines einseitigen Einselementes) nicht eindeutig festgelegt
werden konnen. In diesem Zusammenhang wird noch gezeigt, dass die Automorphismenklassengruppe
der Halbgruppenkategorie die Ordnung 2 hat.
Reviewed by Heinrich Seidel
Previous Review
Next Review
39 #26016.50
Clark,
W. Edwin
Murase's quasi-matrix rings and generalizations.
Sci.
Papers College Gen. Ed. Univ. Tokyo 18 1968 99--109.
For a positive integer $n$, let $QM(n)$ be the semigroup
with elements $e\sb {ij}$, where $1\leq i\leq n$, $i\leq j$ are integers,
together with the zero element 0. Multiplication is defined by $e\sb {hi}e\sb
{jk}=e\sb {h,i-j+k}$ if $i\equiv j (\text{mod}\,n)$ and $e\sb {hi}e\sb
{jk}=0$ otherwise. The contracted semigroup ring $K[QM(n)]$ over a division
ring $K$ is called an infinite quasi-matrix ring of degree $n$ over $K$.
It is shown that every quasi-matrix ring in the sense of I. Murase [same
Papers 13 (1963), 131--158; MR
28
#5086] is a homomorphic image of an infinite quasi-matrix ring and
that every $K[QM(n)]$ is isomorphic to the ring of all $n\times n$ matrices
$(a\sb {ij})$ over the ring of polynomials in $x$ over $K$ such that $x$
divides $a\sb {ij}$ whenever $i>j$. The following principal results are
also proved. (1) $K[QM(n)]$ is prime, left and right Noetherian, and semisimple.
(2) Every proper homomorphic image of $K[QM(n)]$ is Artinian and generalized
uniserial. (3) Every ideal of $K[QM(n)]$ is principal.
Reviewed by J. Luh
Previous Review
Next Review
37 #140516.50
Clark,
W. Edwin
A note on semiprimary ${\rm PP}$-rings.
Osaka
J. Math. 4 1967 177--178.
The author shows that, in a semiprimary ring, principal
left ideals are projective if and only if left annihilators are direct
summands and that these properties also hold with right replacing left.
Reviewed by J. P. Jans
Previous Review
Next Review
36 #645216.50
(18.00)
Clark,
W. Edwin
Algebras of global dimension one with a finite ideal lattice.
Pacific
J. Math. 23 1967 463--471.
Author's introduction: "Let $A$ denote a finite-dimensional
(associative) algebra over an algebraically closed field $K$. In this paper
a specific representation is given for those algebras $A$ which have global
dimension one (or less) and have only a finite number of (two-sided) ideals.
It is shown that every such algebra is isomorphic to a (contracted) semigroup
algebra $K[S]$ over a subsemigroup $S$ of the semigroup of all $n\times
n$ matrix units $\{e\sb {ij}\}\cup\{0\}$ which (i) contains $e\sb {11},\cdots,e\sb
{nn}$ and (ii) contains $e\sb {ij}$ or $e\sb {ji}$ whenever there are $h$
and $k$ such that $e\sb {hi},e\sb {ik}$ and $e\sb {hj},e\sb {jk}$ are in
$S$. Conversely, if $S$ satisfies (i) and (ii) then $K[S]$ has global dimension
one or less and has a finite ideal lattice."
Reviewed by D. S. Rim
Previous Review
Next Review
36 #644716.32
Clark,
W. Edwin
Generalized radical rings.
Canad.
J. Math. 20 1968 88--94.
Let $R$ be a ring and denote by $\circ$ the composition
$a\circ b=a+b-ab$ for $a,b\in R$. It is known that $(R,\circ)$ is a semigroup
and that $R$ is a radical ring in the sense of Jacobson if and only if
$(R,\circ)$ is a group. The author calls $R$ a generalized radical ring
if $(R,\circ)$ is the union of groups. A ring is said to be strongly regular
if $a\in a\sp 2R$ for all $a\in R$. An idempotent of a ring is called principal
if it is an identity for the ring modulo its radical.
The author proves that every strongly regular ring is a generalized
radical ring. It is also shown that a ring $R$ possessing a principal idempotent
is a generalized radical ring if and only it is a splitting extension of
its radical by a strongly regular subring $eRe$ for some idempotent $e\in
R$.
Reviewed by K. G. Wolfson
Previous Review
Next Review
36 #515916.10
Clark,
W. Edwin; Lewin,
Jacques
On minimal ideals in the circle composition semigroup of a ring.
Publ.
Math. Debrecen 14 1967 99--104.
The elements of a ring $R$ form a semigroup $(R,\circ)$
under the circle composition defined by $a\circ b=a+b-ab$. The identity
element of $(R,\circ)$ is the zero element 0 of $R$. There exists the minimum
$K$ of the ideals of $(R,\circ)$, and $K$ is a completely simple ideal
in $(R,\circ)$ if and only if $R$ contains a principal idempotent $e$,
i.e., $e$ is modulo $J$ the identity of $R/J$, where $J$ is the Jacobson
radical of $R$. It is well known that $J$ is the maximum of all such ideals
in the ring $R$ in which every element $a$ possesses an inverse in the
semigroup $(R,\circ)$. If $R$ contains a principal idempotent $e$, then
$K=R\circ e\circ R$ is valid. The ideal of the ring $R$ which is generated
by $K-e$ is denoted by $I(R)$, where $K$ is defined above; $I(R)$ is independent
of the choice of $e$. This ideal is contained in $J$ and is equal to the
subring of $R$ which is generated by the component $P\sb e=(1-e)R+R(1-e)$
in the direct sum (qua abelian groups) $R=eRe+P\sb e$. A linear variety
of an additively written abelian group $A$ consists of $a+M$, where $a\in
A$ and $M$ is a subgroup of $A$. The above subsemigroup $K=R\circ e\circ
R$ of $(R,\circ)$ is a linear variety in the additive group $(R,+)$ of
$R$ if and only if $R=eRe+I(R)$ (direct qua abelian groups) is valid. In
that case $K=e+P\sb e$ and $I(R)=P\sb e$.
Reviewed by E.-A. Behrens
Previous Review
Next Review
35 #547516.55
(20.00)
Clark,
W. Edwin
Twisted matrix units semigroup algebras.
Duke
Math. J. 34 1967 417--423.
The author defines a twisted semigroup algebra as
follows. Let $S$ be a semigroup with zero. Let $F$ be a field and let $\phi\colon
S\times S\rightarrow F$ satisfy (i) $\phi(s,t)=0$ if and only if $st=0$
and (ii) $\phi(r,st)\phi(s,t)=\phi(rs,t)\phi(r,s)$ whenever $rst\neq 0$.
Let $F\sb \phi[S]$ denote the vector space of all formal finite linear
combinations $\sum\alpha\sb is\sb i$ with $\alpha\sb i\in F$ and $s\sb
i\in S$, $s\sb i\neq 0$. Define a product on $F\sb \phi[S]$ by setting
$s·t=\phi(s,t)st$, for $s$ and $t$ non-zero elements of $S$, and
extending linearly to all of $F\sb \phi[S]$. (ii) above insures the associativity
of $F\sb \phi[S]$. $F\sb \phi[S]$ is termed a twisted semigroup algebra
of $S$ over $F$. If $\phi(s,t)=1 [0]$ when $st\neq 0 [st=0]$, then $F\sb
\phi[S]$ is just the ordinary semigroup algebra of $S$ over $F$.
Let $MU(n)$ denote the full semigroup of matrix units $\{e\sb {ij}\colon
1\leq i,j\leq n\}\cup\{0\}$, where $e\sb {hi}e\sb {jk}=\delta\sb {ij}e\sb
{hk}$. By a matrix units semigroup is meant a subsemigroup of $MU(n)$ which
contains $e\sb {11},\cdots,e\sb {nn}$.
The author proves the following structure theorem on finite-dimensional
linear associative algebras, after establishing a number of interesting
results on the characterization of a quasi-Baer, Artinian ring (an Artinian
ring $A$ with unity, such that the left annihilator of every ideal of $A$
is generated, as a left ideal, by an idempotent). Theorem: Let $A$ be a
finite-dimensional algebra over an algebraically closed field $F$. Then,
$A\cong F\sb \phi[S]$ for some matrix units semigroup $S$ if and only if
$A$ is quasi-Baer and has a finite ideal lattice.
Reviewed by L. M. Chawla
Previous Review
Next Review
34 #262116.48
(16.46)
Clark,
W. Edwin
Baer rings which arise from certain transitive graphs.
Duke
Math. J. 33 1966 647--656.
A ring $A$ is called a Baer ring if the left [right]
annihilator of each subset of $A$ is a principal left [right] ideal generated
by an idempotent. Let $D$ be a division ring, and $M\sb n=M\sb n(D)$ be
the ring of all $n\times n$ matrices with entries from $D$. Let $\{e\sb
{ij}\}$ denote the usual $n\sp 2$ matrix units of $M\sb n$. The set of
all matrix units together with the zero matrix constitute the semigroup
$MU(n)$. If $S$ is any subsemigroup of $MU(n)$, the set $A(S,D)$ of all
linear combinations over $D$ of elements of $S$ forms a ring. Now let $S$
be any subsemigroup of $MU(n)$ containing $e\sb {ii}$ for $i=1,2,3,\cdots,n$.
The main theorem of the paper then asserts that $A(S,D)$ is a Baer ring
if and only if $S$ also satisfies the following condition: $(*)$ if $h,i,j,k$
are distinct integers such that $e\sb {ki},e\sb {ih},e\sb {kj}$, and $e\sb
{jh}\in S$, then $e\sb {ij}$ or $e\sb {ji}\in S$. This generalizes the
known result that the complete ring of triangular matrices over a division
ring is a Baer ring.
The title of the paper derives from the fact that there is a one-to-one
correspondence between subsemigroups $S$ of $MU(n)$ containing $e\sb {11},e\sb
{22},\cdots,e\sb {nn}$ that satisfy $(*)$ and certain transitive directed
graphs on $n$ vertices. Actually no graph theory is involved in the arguments.
Reviewed by K. G. Wolfson
Previous Review
Next Review
32 #765920.92
(16.44)
Clark,
W. Edwin
Weakly semi-simple finite-dimensional algebras.
Canad.
J. Math. 18 1966 433--442.
Let $B$ be a finite-dimensional algebra over a field
$F$. If $\varphi\colon B\rightarrow F$ is an epimorphism (i.e., $(B,\varphi)$
is a supplemented $F$-algebra in the sense of Cartan and Eilenberg [Homological
algebra, Princeton Univ. Press, Princeton, N.J., 1956; MR
17, 1040]), then the multiplicative semigroup $S=\varphi\sp {-1}(1)$
is called a translate of the algebra $A=\text{Ker}\,\varphi$. The author
has shown [\#7658 above] that $S$ has a kernel $K$ (= unique minimal ideal
of $S$) which is a union of subgroups. Let $R(S)$ be the radical of the
semigroup (i.e., union of all ideals $I$ with $I\sp n\subseteq K$) and
let $R(A)$ be the radical of the algebra $A$. It is shown here that for
each $e=e\sp 2\in K$, $R(S)=R(A)+e$, whence $A$ is nilpotent if and only
if $R(S)=S$. The algebras of the title deal with the other extreme: a finite-dimensional
algebra $A$ is weakly semi-simple if it has a translate $S$ with $R(S)=K$.
For such an $A$, if $f$ is a principal idempotent, then (i) $fAf$ is semi-simple,
and (ii) $A=fAf\oplus\,R(A)$ as a vector space. Conversely, a principal
idempotent satisfying (i) or (ii) implies weak semi-simplicity. Algebras
of class $Q$ [Thrall, Canad. J. Math. 7 (1955), 382--390; MR
16, 992] are precisely the weakly semi-simple algebras $A$ with $A\sp
2=A$ and $R(A)\sp 3=0$.
Reviewed by F. W. Anderson
Previous Review
Next Review
32 #765820.92
Clark,
W. Edwin
Affine semigroups over an arbitrary field.
Proc.
Glasgow Math. Assoc. 7 1965 80--92 (1965).
An affine semigroup $S$ is a linear variety (i.e.,
a translate of a subspace of a vector space) endowed with an associative
multiplication for which the mappings $x\rightarrow xa$ and $x\rightarrow
ax (x\in S)$ are affine mappings for all $a\in S$. The affine semigroups
of Haskell Cohen and H. S. Collins [Trans. Amer. Math. Soc. 93 (1959),
97--113; MR 21
#6400] are shown to be subsemigroups of such affine semigroups. Affine
semigroups are characterized as follows: They are semigroups $S$ isomorphic
to semigroups $\phi\sp {-1}(1)$, where $\phi$ is a $\Phi$-epimorphism of
an algebra $A$ over $\Phi$ onto the field $\Phi$ and $\phi\sp {-1}(1)$
is regarded as a subsemigroup of the multiplicative semigroup of $A$. If
$S$ is finite-dimensional, then it can be represented faithfully as a (multiplicative)
semigroup of matrices over $\Phi$ (the vector space involved being that
spanned by the matrices), and it can be decomposed, by an appropriate partition
of the matrices, into a group, two vector spaces, and an algebra. The kernel
$K$, which then always exists, of $S$ may be characterized in terms of
such a decomposition. Let $M(K)$ denote the linear variety generated by
$K$. Then $M(K)\sp n=K$, if $n$ is greater than the dimension of $M(K)$.
If the characteristic of $\Phi$ is different from 2, if every element of
$K$ is idempotent, and if $K$ is not a variety, then $M(K)$ contains a
subsemigroup isomorphic to the semigroup [cf. Cohen and Collins, loc. cit.]
$\Phi\oplus\Phi\oplus\Phi$, with multiplication $(x,y,z)(a,b,c)=(x,b,ab)$,
where $xb$ denotes multiplication in $\Phi$.
Reviewed by G. B. Preston
Previous Review
Next Review
31 #131120.92
Clark,
W. Edwin
Remarks on the kernel of a matrix semigroup. (English. Russian summary)
Czechoslovak
Math. J. 15 (90) 1965 305--310.
If $S$ is a matrix semigroup (matrices of finite order
over a field), denote by $m(S)$ the set of all elements of $S$ with minimal
rank. If $S$ contains a kernel $K$, then $K\subseteqq m(S)$.
An example is given which shows that $m(S)$ need not be equal to $K$.
The same example gives a negative answer to a question which has been open
for some time: If $K$ is the kernel of a semigroup $S$, is a left ideal
of $K$ necessarily a left ideal of $S$?
Call a semigroup $S$ pseudo-invertible if for any $a\in S$ there is
a power $a\sp n$ which is contained in a subgroup of $S$. It is proved
that for a pseudo-invertible semigroup of matrices, $m(S)=K$ and $K$ is
completely simple. Conversely, for any matrix semigroup $S$ for which $K$
is completely simple, $K=m(S)$.
Reviewed by St. Schwarz
Previous Review
CMP 1 369 267(96:06) 05C35
Clark,
W. Edwin(1-SFL);
Shekhtman,
Boris(1-SFL)
On the domination matrices of the ${\scr C}$-analogues of Kneser
graphs. (English. English summary)
Proceedings of the Twenty-sixth Southeastern International Conference
on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1995).
Congr.
Numer. 107
(1995),
193--197.
{There will be no review of this item.}
Previous Review
Return
to List
(c) 2001, American Mathematical Society