Homomorphisms and subdirect decompositions of semi-groups

On subdirectly irreducible commutative semigroups

“…In the elementary case, our two characterizations and construction from groups and nilsemigroups are still new. In particular they go far deeper than Schein's results in [14], as one could expect since our results are also considerably less general.…”

“…The proofs of our results depend on techniques of congruence construction which were essentially developed by Schein in [14].…”

Subdirectly irreducible idempotent semigroups

“…REMARK 2. Special cases of the above results are known ( [2], [3], [4], [7]) In particular the paper of M. Petrich [3] gives the results of our next section. We include the next section, however, for completeness and because the arguments become especially simple in view of Theorem 2.4 and its Corollary.…”

The syntactic monoid of an infix code

“…Remarks. In the presence of condition (ô), according to [3,Theorem 3.7], condition (e) is equivalent to M being subdirectly irreducible. Even though this latter notion is a more familiar one, the existence of a core is a condition easier to verify.…”