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Coherence in varieties of algebras. (English) Zbl 0704.08003

An algebra A has coherent subalgebras if for any subalgebra B of A and any congruence \(\Theta\) on A, \([b]\Theta\subseteq B\) for some \(b\in B\) implies \([x]\Theta\subseteq B\) for all \(x\in B\). A variety V has coherent subalgebras whenever each V-algebra has this property. Mal’cev conditions for a variety with coherent subalgebras were given by D. Geiger [Am. Math. Soc., Notices 21, A-436 (1974)]. This paper gives such conditions for varieties with coherent congruence blocks which have an analogous definition. The main theorem is:
Theorem. For any variety V the following conditions are equivalent:
(1) V has coherent congruence blocks;
(2) there exist an integer n, ternary terms \(b_ 1,...,b_ n\) and ternary terms \(s_ 1,...,s_ n\) such that the identities
\(x=s_ 1(x,y,b_ 1(x,y,x))\),
\(s_ i(x,y,x)=s_{i+1}(x,y,b_{i+1}(x,y,x))\), \(1\leq i<n,\)
\(y=s_ n(x,y,x)\), and
\(z=b_ i(x,x,z)\), \(1\leq i\leq n,\)
hold in V;
(3) there exist an integer n and ternary terms \(b_ 1,...,b_ n\) such that
\(z=b_ i(x,x,z)\), \(1\leq i\leq n\), and
\((x=b_ i(x,y,x)\), \(1\leq i\leq n)\to x=y\)
hold in V.
Various properties of the congruence lattices of algebras in varieties with coherent congruence blocks are valid as corollaries.
[Reviewer’s remark: Corollary 1, Any variety with coherent subalgebras is permutable, attributed to Geiger, does not seem to be in the abstract quoted; nor do I find the proof given convincing.]
Reviewer: S.Oates-Williams

MSC:

08B05 Equational logic, Mal’tsev conditions
08B10 Congruence modularity, congruence distributivity
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References:

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