Near-Rings

Definition

• x + (y + z) = (x + y) + z,
• -x + x =x+ -x = 0, 0 + x =x+0= x,
• x(yz) = (xy)z,
• (x+y)z = xz + yz.

Or more simply an algebra (N,+,*) such that

• (N,+) is a (not necessarily abelian) group
• (N.*) is a semigroup
• left distributivity (a+b)*c = a*c+b*c holds.

Examples

• Any ring is also a near-ring.
• The mappings of a group into itself under pointwise addition and composition of maps.

Structure

Representation

Every nearring is embeddable into a nearring of mappings as in the second example above.

Decision problems

Identity problem: Solvable, there exists a critical pair complete rewriting system deciding the free word problem for nearrings. (E. Aichinger 1994)

Word problem: Unsolvable. If it were solvable, the group problem would be solvable, and it isn't.

Spectra and growth

Finite spectrum:Recent work by Christof Nöbauer has found the following Spectrum (numbers of non-isomorphic nearrings defineable on a given additive group) for nearrings of low order:

• C_2: 3
• C_3: 5
• C_4: 12
• V_4: 23
• C_5: 10
• C_6: 60
• S_3: 39
• C_7: 24
• C_8: 135
• C_2 x C_4: 1159
• C_2 x C_2 x C_2: 834
• D_8: 1447
• Q_8: 281
• C_9: 222
• C_3 x C_3: 264
• C_10: 329
• D_10: 206
• C_11: 1312
• C_12: 1749
• C_2xC_6: 3501
• D_12: 48137
• A_4: 482
• T: 824
• C_13: 454
• C_14 2716
• D_14: 1821
• C_15: 3817

Free spectrum:

Growth series:

History/Importance

Generalisation of nearfields, dynamical systems. See here for more.

References

Nearring homepage

G Pilz "Nearrings" North-Holland mathematics Studies 23, "nd ed. 1983

J.D.P. Meldrum, "Near-rings and their links with groups," Research Notes in Mathematics vol. 134, Pitman, London, 1985.

J. R. Clay "Nearrings: Geneses and Applications" Oxford 1992.

Subsystems

Distributive nearrings, distributiely generated nearrings, commutative nearrings, abelian nearrings, nearfields, planar nearrings.

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A Catalogue of Algebraic Systems / John Pedersen / jfp@math.usf.edu
This entry by Tim Boykett