\(LC\)-commutative permutable semigroups (Q1910645)

A semigroup \(S\) is called permutable if \(\rho \circ \sigma = \sigma \circ \rho\) for all congruences \(\rho\), \(\sigma\) on \(S\). A semigroup is called \(L\)-commutative if for every \(a, b \in S\) there is an element \(x \in S^1\) such that \(ab = xba\). \(S\) is conditionally commutative if, for any \(a, b \in S\), \(ab = ba\) implies \(axb = bxa\) for all \(x\in S\). An \(L\)-commutative conditionally commutative semigroup is called \(LC\)-commutative. In this paper the structure of \(LC\)-commutative permutable semigroups is completely determined as follows: I. \(LC\)-commutative permutable archimedean semigroups. (1) \(S\) is a direct product of an abelian group \(G\) and a left zero semigroup \(L\) with \(|L|\leq 2\) or (2) \(S\) is a commutative nilsemigroup whose ideals form a chain with respect to inclusion. II. \(LC\)-commutative permutable nonarchimedean semigroups. (1) \(S\) is a commutative semigroup which is an ideal extension of a nil semigroup \(W\) by an abelian group \(G\) with zero such that the orbits of \(W\) under the action by \(G\) is a naturally totally ordered semigroup or (2) \(S\) is constructed by \(T\) of the form (1) and an ideal \(J\) of \(T\), the so-called bifurcate extension of \(T\) by \(J\). Finally the author discusses \(LC\)-commutative \(\Delta\)-semigroups where a \(\Delta\)-semigroup is a semigroup the lattice of congruences of which forms a chain.