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Proper two-sided restriction semigroups and partial actions. (English) Zbl 1258.20047
An algebra \((S,\cdot,+,*)\) of type \((2,1,1)\) is called a (two-sided) restriction semigroup if \((S,\cdot )\) is a semigroup, \((S,\cdot,+)\) satisfies the identities \(x^+x=x\), \(x^+y^+=y^+x^+\), \((x^+y^+)^+=x^+y^+\), \(xy^+=(xy)^+x\), \((S,\cdot,*)\) satisfies the dual identities, and \((x^+)^*=x^+\), \((x^*)^+=x^*\).
The structure of restriction semigroups is studied. Using both sides actions of a monoid on a semilattice, a proper restriction semigroup is constructed. To obtain the full class of proper restriction semigroups, partial actions of monoids are used. Thus the main result is a structure theorem for proper restriction semigroups.
Reviewer’s comment: The topic of the paper under review is close to that of a paper by M. B. Szendrei, [Int. J. Algebra Comput. 22, No. 3, 1250024 (2012; Zbl 1258.20048)].

MSC:
20M10 General structure theory for semigroups
20M30 Representation of semigroups; actions of semigroups on sets
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