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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

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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