## 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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