Three examples of join computations (Q1266926)

Let \(\mathbf G\) be the pseudovariety of finite groups, \(\mathbf D\) the pseudovariety of semigroups in which each idempotent is a right zero and \(\mathbf{MK}\) the pseudovariety generated by monoids \(M\) such that each idempotent of \(M\setminus\{1\}\) is a left zero. \textit{J. Almeida} [see Problem 24 in: Finite semigroups and universal algebra, World Scientific (1994; Zbl 0844.20039)] proposed to calculate the pseudovariety joins \(\mathbf{MK}\vee{\mathbf D}\), \(\mathbf{MK}\vee{\mathbf G}\) and \(\mathbf{MK}\vee{\mathbf D}\vee{\mathbf G}\). The authors solve the problem by finding a finite pseudoidentity basis for each of the joins. Namely, \(\mathbf{MK}\vee{\mathbf D}\) is shown to be defined by the pseudoidentities (1) \(x^\omega yx^\omega zt^\omega=x^\omega yzt^\omega\) and (2) \(x^\omega=x^{\omega+1}\) (as it was conjectured by Almeida), \(\mathbf{MK}\vee{\mathbf G}\) is defined by the pseudoidentities (3) \((xy^{\omega+1}z)^\omega=(xy^\omega z)^\omega\) and (4) \(x^\omega yx^\omega=x^\omega y\), and \(\mathbf{MK}\vee{\mathbf D}\vee{\mathbf G}\) is defined by (1) and (3).