Finite Semigroups with Infinite Irredundant Identity Bases

A basis of identities Σ for a semigroup S is a set of identities satisfied by S from which all other identities of S can be derived. The basis Σ is said to be irredundant (or irreducible) if no proper subset of Σ is a basis for the identities of S. If the basis Σ is finite, then it is always possible to extract an irredundant basis, however if Σ is infinite then it is conceivable that no irredundant basis exists. According to [20], initial optimism led to the supposition that all finite semigroups without a finite basis of identities might at least have an irredundant basis of identities. Subsequent examples of finite semigroups without irredundant identity bases (see [8], [9] or [14] for example) have shown this to be false, and moreover have provided increasing evidence that there are no finite semigroups with infinite irredundant bases of identities. The possible existence of such a semigroup was first explicitly raised as far back as [16, Question 2.51a] and then in [17, Question 8.6] and most recently in [20, Problem 2.6], where it is speculated that the answer might be negative. We show that the answer is in fact positive and, at least within a restricted class, there are numerous examples. The main result of the paper is Theorem 2.6 of Sec. 2, with our smallest example established in Proposition 2.7. In Sec. 3 we extend the applicability of this theorem. The proofs up to this point are not constructive in any practical sense and so Sec. 4 is devoted to giving a finite semigroup with an infinite irredundant basis that is