On a class of completely join prime \(J\)-trivial semigroups with unique involution (Q2411685)

Here, series of \(J\)-trivial semigroups \({L_l} = \left\langle {E,F\mid {E^2} = E,{F^2} = F,\underbrace {EFEFEF\dots}_l = 0} \right\rangle \), \(l \geqslant 2\), of order \(2l\) having a unique involution \(*\) are investigated. It is shown, that the infinite chain \(\mathsf{Var}\,{L_2} \subset \mathsf{Var}\,{L_3} \subset \mathsf{Var}\,{L_4} \subset \cdots\) in the lattice of pseudovarieties of semigroups is incompatible with infinite chains \(\mathsf{Var}({L_2},*) \subset \mathsf{Var}({L_4},*) \subset \mathsf{Var}({L_6},*) \subset \cdots\), \(\mathsf{Var}({L_3},*) \subset \mathsf{Var}({L_5},*) \subset \mathsf{Var}({L_7},*) \subset \cdots\) in the lattice of pseudovarieties of semigroups with involution. Thus, these semigroups are an example of series of semigroups with unique involution such that in the lattice of pseudovarieties of semigroups \(\mathsf{Var}\,{L_l} \subset \mathsf{Var}\,{L_{l + 1}}\), but in the lattice of pseudovarieties of semigroups with involution \(\mathsf{Var}({L_l},*) \not\subset \mathsf{Var}({L_{l + 1}},*)\). The algebras \({L_l}\) and \(({L_l},*)\) are completely join prime (also called join irreducible), i.e., if one of them belongs to the complete join of some collection of pseudovarieties then it belongs to one of the pseudovarieties of the collection.