Finite basis problem for involution monoids of unitriangular boolean matrices

Tropical representations and identities of the stylic monoid

“…By [44,Theorem 4], we have that any twisted involution semigroup whose semigroup reduct is non-finitely based must also be non-finitely based. Since styl n is non-finitely based for n ≥ 4, by Corollary 4.2, the following is immediate: Proof As a consequence of Proposition 5.1, (styl 2 , *) satisfies condition (4A) of [64,Theorem 4.1]. As such, we only need to prove that (styl 2 , *) satisfies condition (4B) to show that it is not finitely based.…”

“…Notice that [64, Lemma 4.9] holds for (styl 2 , *): By the same reasoning as the one given in the proof of [64,Lemma 4.9], and replacing the evaluation used in that proof by any evaluation ϕ : X → styl 2 such that ϕ(x) = [1] styl 2 gives us the result. Furthermore, [64,Lemma 4.10] also holds for (styl 2 , *), by taking any evaluation ϕ : X → styl 2 which maps x and y to [1] styl 2 . Thus, (styl 2 , *) satisfies condition (4B), and is therefore non-finitely based.…”

“…Finally, we look at the case of (styl 3 , *): Proof As a consequence of Proposition 5.1, (styl 3 , *) satisfies condition (5A) of [64,Theorem 5.2]. As such, we only need to prove that (styl 3 , *) satisfies conditions (5B) and (5C) to show that it is not finitely based.…”

Minimal generating sets for matrix monoids

“…The monoid M n (B) has several natural submonoids, such as the monoids: M id n (B) consisting of the reflexive boolean matrices (that is matrices containing the identity matrix); M S n (B) consisting of all Hall matrices (matrices containing a permutation); and U T n (B) of upper triangular boolean matrices. Each of these submonoids has been extensively studied in their own right; see, for example, [8,11,22,36,39,49,54,58,59,62]. Unlike M n (B), M id n (B) is J -trivial, and so has precisely |M id n (B)| = 2 n 2 −n J -classes.…”

“…. The monoid U T n (B) appears to have been primarily studied in the context of varieties; see for instance, [39,62]. It is somewhat surprising that there appears to be no description of the unique minimal generating set of U T n (B) in the literature, in particular because this minimal generating set is more straightforward to determine than that of the other submonoids of M n (B) we consider (see Section 3.5).…”