A minimal nonfinitely based semigroup whose variety is polynomially recognizable

Abstract

We exhibit a 6-element semigroup that has no finite identity basis but nevertheless generates a variety whose finite membership problem admits a polynomial algorithm.

Flexible constraint satisfiability and a problem in semigroup theory

“…The six element monoid B 1 2 = 1, a, b | a 2 = b 2 = 0, aba = a, bab = b is the most ubiquitous minimal counterexample in problems relation to varieties of semigroups. The possible intractability of the computational problem of deciding membership of finite semigroups in the variety generated by B 1 2 is perhaps the most obvious unresolved problem relating to B 1 2 and so it is not surprising that this has appeared in a number of places in the literature including Problem 4 of Almeida [1, p. 441], Problem 3.11 of Kharlampovich and Sapir [23] and page 849 of Volkov, Gol ′ dberg and Kublanovksiȋ [38]. We now use the results of the previous section to show that this computational problem is NP-hard.…”

“…Following Lee and Zhang [29] it is known that there are precisely 4 semigroups of order 6 that have no finite basis for their equations. Aside from B 1 2 , which we have shown to generate a pseudovariety with hard membership problem, and A 1 2 which we have discussed in Problem 7.5, there is the example A g 2 observed by Volkov (private communication; see [38]) and the semigroup L of Zhang and Luo [40]. The semigroup A g 2 was shown to have polynomial time membership problem for its pseudovariety by Goldberg, Kublanovsky and Volkov [38], but the complexity of membership in V(L) is unknown.…”