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Proper left type-$$A$$ monoids revisited. (English) Zbl 0802.20051
The relation $${\mathcal R}^*$$ on a semigroup $$S$$ is defined by $$a{\mathcal R}^* b$$ if $$a {\mathcal R}b$$ in some oversemigroup of $$S$$. $$S$$ is an $$E$$- semigroup if its set $$E(S)$$ of idempotents is a subsemilattice of $$S$$. An $$E$$-semigroup is left adequate if every $${\mathcal R}^*$$-class contains an idempotent. This idempotent is unique and denoted for $$a \in S$$ by $$a^ +$$. A left adequate semigroup is left type-$$A$$ if $$ae = (ae)^ + a$$ for each $$a \in S$$ and $$e \in E(S)$$. On a left type-$$A$$ semigroup there is a minimum right cancellative congruence $$\sigma$$. A left type-$$A$$ semigroup is called proper if $$\sigma \cap {\mathcal R}^* = \iota$$. Several characterizations of proper left type-$$A$$ monoids and $$E$$-unitary left type-$$A$$ monoids are presented. For instance, every $$E$$-unitary (resp. proper) left type $$A$$ monoid is isomorphic to a monoid $$C_ u$$, where $$u$$ is an object of a (proper) idempotent left type-$$A$$ category on which a right cancellative monoid $$T$$ acts, and $$C_ u = \{(t,p)\mid t\in T,\;p\in \text{Mor}(ut,u)\}$$ and multiplication is defined by $$(t,p)(h,q) = (th,ph + q)$$.
Reviewer: J.Henno (Tallinn)

##### MSC:
 20M10 General structure theory for semigroups 20M50 Connections of semigroups with homological algebra and category theory 20M15 Mappings of semigroups
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##### References:
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