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The lattice of varieties of symmetric idempotent entropic groupoids. (English) Zbl 0663.20065
An algebra $$(G,\cdot)$$ with a binary operation $$\cdot$$ satisfying each of the identities $$(x\cdot y)\cdot y=x$$, $$x\cdot x=x$$, $$(x\cdot y)\cdot (z\cdot t)=(x\cdot z)\cdot (y\cdot t)$$ is called a symmetric idempotent entropic groupoid (SIE-groupoid). The aim of this paper is to describe the lattice L(SIE) of all subvarieties of the variety SIE of all SIE- groupoids. Let L be a lattice. By $$L^+$$ is denoted the lattice $$L\cup \{\infty \}$$ in which the element $$\infty$$ is greater than all elements of $$L$$. Let $$N$$ denote the lattice of all natural numbers with respect to the partial order of divisibility. At last let $$F(x,y)$$ be the free SIE- groupoid on two generators $$x$$ and $$y$$, $$w_ 0(x,y)=x$$, $$w_ 1(x,y)=y$$ and $$w_ i(x,y)=w_{i-2}(x,y)\cdot w_{i-1}(x,y)$$ for $$i\geq 2.$$
The main result of this paper is the following: Theorem 5.3. The lattice $$L$$(SIE) is isomorphic to $$N^+$$. The variety corresponding to $$\infty$$ is SIE. If $$n$$ is a natural number then the variety corresponding to $$n$$ is just the variety of SIE-groupoids satisfying the identity $$w_ n(x,y)=x$$.
Reviewer: L.Martynov

##### MSC:
 20M07 Varieties and pseudovarieties of semigroups 08B15 Lattices of varieties 20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
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