INTERPRETING GRAPH COLORABILITY IN FINITE SEMIGROUPS

MARCEL JACKSON, RALPH McKENZIE
2006 International journal of algebra and computation  
We show that a number of natural membership problems for classes associated with finite semigroups are computationally difficult. In particular, we construct a 55-element semigroup S such that the finite membership problem for the variety of semigroups generated by S interprets the graph 3-colorability problem. Recall that the term-equivalence problem for an algebra A is the problem of deciding for two terms s, t in the signature of A, if A |= s ≈ t. This problem is known to be in co-NP and
more » ... e are now a number of known semigroups for which this problem is co-NP-complete (such as B 1 2 ; see [14] and [8] ). If V is a variety with a finite basis of equations, then the finite membership problem for V can be solved in polynomial time (simply test for satisfaction of the finite set of equations). Assuming that P = NP, it follows that the semigroup variety generated by S(C 3 ) is not finitely based. The same holds in the monoid case and in this case the absence of a finite equational basis is easily established using the fact that for any binary relational structure G, the monoid S 1 (G) contains a submonoid isomorphic to the inherently non-finitely based semigroup B 1 2 (see [13] ) and hence has no finite equational basis itself. Results of [13] also show that the semigroup S(G) is never inherently nonfinitely based. However, we will show that in most cases S(G) is non-finitely based. Theorem 4.1. Let G be a graph with finite chromatic number that is not a disjoint union of complete bipartite graphs. Then S(G) is not finitely based.
doi:10.1142/s0218196706002846 fatcat:tl2xcqfsnvcxvg62nwxycksepq