The join of the pseudovarieties of R-trivial and L-trivial monoids

The join of the pseudovarieties of idempotent semigroups and locally trivial semigroups

“…Reiterman's theorem [29] is the starting point of the equational theory of pseudovarieties: it states that pseudovarieties are defined by pseudoidentities, just as varieties are defined by identities. Using this theorem and some ad hoc facts, Almeida, Azevedo and Weil [3,2,8,10,11,14,13] computed some non-trivial joins for which algebraic methods failed. -Sometimes, one can only determine whether the join has a finite basis of pseudoidentities.…”

Pointlike sets with respect to R and J

“…Using the compactness of Ω A S, continuity of the content function, and the fact that Ω A S is dense in Ω A S, it is easy to show that every non-empty pseudoword admits at least one left basic factorization. The following result from [6] is the fundamental observation for the identification of pseudowords over R.…”

“…By a result of the first author and Azevedo [6] (see [2, Theorem 9.2.13]), there is no pseudoidentity valid in R ∨ L in which one side belongs to ψ −1 • ψ(abc) and the other to ψ −1 • ψ(ab 2 c), and so the set {abc, ab 2 c} is not pointlike with respect to the relational morphism µ R∨L .…”

Complete reducibility of systems of equations with respect to R

“…Yet, there are many common examples of weakly cancellable pseudovarieties such as R [9] and J (finite J-trivial semigroups) [2]. A stronger requirement is that V be closed under Birget expansions [26], 4 which is the case for any pseudovariety of the form V = B m W [17], where B is the pseudovariety of all finite bands (semigroups in which all elements are idempotents), which in turn is equivalent to V = B m V. Using these results, one may show that pseudovarieties such as OCR (finite orthodox completely regular semigroups), CR (finite complete regular semigroups), H (finite semigroups whose subgroups lie in a given pseudovariety of groups H), DA (finite semigroups in which regular elements are idempotents), DO (finite semigroups in which regular D-classes are orthodox subsemigroups), DS (finite semigroups in which regular D-classes are subsemigroups), as well as any meet or join of some of these pseudovarieties, are weakly cancellable [17].…”