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Semigroups and their natural order. (English) Zbl 0816.20057
For any semigroup \(S\) the relation defined by \(a \leq b\) if and only if \(a = xb = by\), \(xa = a\) for some \(x,y \in S^ 1\) is called the natural partial order on \(S\) and is known to generalize the well-known relation of the same name for regular semigroups. The paper addresses and partially answers the general questions: When is \(\leq\) trivial? When is it total? When is it compatible with multiplication? Applications are given concerning least primitive congruences, retract extensions of regular semigroups, and strong semilattices of semigroups of certain types. Along the way, an extension of the corollary of Green’s Lemma on the equipotency of \(\mathcal H\)-classes within a \(\mathcal D\)-class is proved, extending the result to the corresponding order ideals of such elements.

20M10 General structure theory for semigroups
06F05 Ordered semigroups and monoids
20M15 Mappings of semigroups
20M17 Regular semigroups
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