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Varieties of left distributive semigroups. (English) Zbl 0589.20037

A semigroup satisfying the identity \(xyz=xyxz\) (resp. \(zyx=zxyx)\) is said to be left (resp. right) distributive. We denote by L the variety of left distributive semigroups. Throughout the paper, let W be a free semigroup over an infinite set X of variables. For r,s\(\in W\), let \(Mod(r=s)\) designate the variety of semigroups satisfying the identity \(r=s\) and put \(M(r=s)=L\cap Mod(r=s)\). Further, we denote by o(r) and (r)o the first and the last variable occurring in r and by var(r) the set of variables contained in r. We put \(l(x)=1\) for every \(x\in X\) and \(l(rs)=l(r)+l(s)\). Let S be a semigroup. Then the relations p(S) and q(S) defined by (a,b)\(\in p(S)\) and (c,d)\(\in q(S)\) iff \(ae=be\) and \(ec=ed\) for every \(e\in S\) are congruences of S.

MSC:

20M07 Varieties and pseudovarieties of semigroups
20M05 Free semigroups, generators and relations, word problems
20M10 General structure theory for semigroups
20M15 Mappings of semigroups
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