Relatively Inherently Nonfinitely Q-based Semigroups

We prove that every semigroup S whose quasivariety contains a 3-nilpotent semigroup or a semigroup of index more than 2 has no finite basis for its quasi-identities provided that one of the following properties holds: • S is finite; • S has a faithful representation by injective partial maps on a set; • S has a faithful representation by order preserving maps on a chain. As a corollary it is shown that, in an asymptotic sense, almost all finite semigroups and finite monoids admit no finite basis for their quasi-identities. Background and motivation Recall that a semigroup quasi-identity is a sentence (∀x) [(p1(x) ≈ q1(x) & · · · & pn(x) ≈ qn(x)) → p(x) ≈ q(x)] where n is nonnegative integer and each pi(x) and qi(x) are semigroup words in the variables x = x1, . . . , xm. Note that a semigroup identity (∀x) (p(x) ≈ q(x)) is a semigroup quasi-identity; this corresponds to the situation where n = 0. We adopt the usual convention of omitting the universal quantifiers from quasiidentities. Standard examples of semigroup quasi-identities are the cancellation laws xy ≈ xz → y ≈ z and yx ≈ zx → y ≈ z; however there are many other familiar properties described by quasi-identities. For example, the statement that idempotents related by Green’s J -relation cannot form a nontrivial subsemilattice can be written as xey ≈ f & ufv ≈ e & e ≈ e & f ≈ f & ef ≈ e & fe ≈ e → e ≈ f. This quasi-identity is satisfied by all finite semigroups, but fails, say, on the bicyclic semigroup. Received by the editors June 4, 2007. 2000 Mathematics Subject Classification. Primary 08C15, 20M20.