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.


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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