Semigroups of order-decreasing graph endomorphisms (Q1279802)

This paper studies the varieties (meaning here pseudovarieties of finite semigroups) and languages generated by semigroups of order-decreasing graph endomorphisms on a (finite) poset. Links with related conditions are also investigated: for instance, a mapping is a graph endomorphism on a chain with loops at each point if and only if it is a contraction mapping. Some sample results are that the variety \(GE\) generated by all semigroups of order-decreasing graph endomorphisms on a chain equals \(MK\), the variety generated by the monoids \(S^1\), where \(S\in K\), the variety of semigroups whose idempotents are left zeros. The class of \(\mathcal R\)-trivial semigroups that have the property that left multiplication by an idempotent is a homomorphism is a variety strictly containing \(GE\) (though both varieties have the same monoids) but strictly contained in the variety generated by partial order-decreasing mappings on a chain which, in turn, is a proper subvariety of \(\mathcal R\)-trivial semigroups. The languages corresponding to the variety \(GE\) are explicitly described.