Semilattices of bisimple orthodox semigroups

A structure theorem for bisimple orthodox semigroups was given by Clifford [2]. In this paper we determine all homomorphisms of a certain type from one bisimple orthodox semigroup into another, and apply the results to give a structure theorem for any semilattice of bisimple orthodox semigroups with identity in which the set of identity elements forms a subsemigroup. A special case of these results is indicated for bisimple left unipotent semigroups.     

Semilattices of bisimple inverse semigroups

The purpose of this paper is to give a structure for a semigroup which is a semilattice of bisimple inverse semigroups and satisfies certain conditions. For such a semigroup, we characterize the idempotent separating congruences.     

Matrices of bisimple regular semigroups

A semigroup S is a matrix of subsemigroups S_iμ, i ε I, μ ε M if the S_iμ form a partition of S and S_iμS_jν≤S_iν for all i, j in I, μ, ν in M. If all the S_iμ are bisimple regular semigroups, then S is a bisimple regular semigroup. Properties of S are considered when the S_iμ are bisimple and regular; for example, if S is orthodox then each element of S has an inverse in every component S_iμ.     

Some congruences on trioids

We present the least idempotent congruence on the trioid with a commutative operation, the least semilattice congruence on the trioid with an idempotent operation, and the least separative congruence on the trioid with a commutative operation. Also we construct different examples of trioids.     

Compact factor congruences imply Boolean factor congruences

We prove that any variety in which every factor congruence is compact has Boolean factor congruences, i.e., for all A in the set of factor congruences of A is a distributive sublattice of the congruence lattice of A.