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On some groupoid modes. (English) Zbl 0669.08005
In Algebra Univ. 1, 73-79 (1971; Zbl 0219.08006), J. Płonka has shown that there are exactly four types of algebras having exactly n n- ary operations depending on all their variables for each $$n=1,2,..$$. One of these types form so-called 2-cyclic groupoids satisfying the following axioms: (L) $$(x\cdot y)\cdot z=(x\cdot z)\cdot y$$, (I) $$x\cdot x=x$$, (R) $$x\cdot (y\cdot z)=x\cdot y$$, $$(C_ 2)$$ $$(x\cdot y)\cdot y=x$$. More general, k-cyclic groupoids satisfy the axioms (L), (I), (R) and $$(C_ k)$$ $$(...((x\cdot y)\cdot y)...)\cdot y=x.$$
In this paper we study more general LIR-groupoids that satisfy only the axioms (L), (I) and (R). In Section 1 we show that they are modes, i.e. they are idempotent and entropic as defined by the first author and J. D. H. Smith [Modal theory. An algebraic approach to order, geometry, and convexity (1985; Zbl 0553.08001)]. So our investigations belong to the recently quickly developed theory of groupoids modes. In Section 2 we prove that a groupoid is an LIR-groupoid if and only if it can be constructed by means of a construction we described earlier, that generalizes a similar construction given by Płonka [loc. cit.] for k- cyclic groupoids. Section 3 is devoted to free LIR-groupoids and Section 4 to properties of identities satisfied in LIR-groupoids. Finally, in Section 4 we describe the lattice of all varieties of LIR-groupoids.

##### MSC:
 08B15 Lattices of varieties 20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms) 08B20 Free algebras
##### Keywords:
LIR-groupoids; groupoids modes