Mal’cev functions on smalgebras. (English) Zbl 0993.08007

Let \(A\) be a nonvoid set. A function \(p:A\times A\times A\to A\) is called a Mal’tsev function on \(A\) whenever \(p(x,y,y)= p(y,y,x)=x\) holds for all \(x,y\in A\). The authors show that for \(|A|=9\) and a lattice \(L\) of permuting equivalences on \(A\) there is a Mal’tsev function on \(A\) that preserves all members of \(L\). The same statement for single algebras with a limited number of elements (= smalgebras) was previously known to hold for \(|A|\leq 8\) and to fail for \(|A|\geq 25\). The problem remains open for \(10\leq |A|\leq 24\).


08B05 Equational logic, Mal’tsev conditions
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