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Finite lattices as lattices of relative congruences of finite unars and Abelian groups. (Russian, English) Zbl 1015.06003

Algebra Logika 40, No. 3, 302-308 (2001); translation in Algebra Logic 40, No. 3, 166-169 (2001).
For any quasivariety \(R\) of algebras and an algebra \(A\), a congruence \(\theta\in \operatorname{Con}A\) of \(A\) is an \(R\)-congruence if the factor \(A/\theta\) belongs to \(R\). The set of all \(R\)-congruences \(\operatorname{Con}_RA\) of \(A\) is an algebraic lattice with respect to the inclusion relation. It is called the lattice of \(R\)-congruences of \(A\).
A lattice \(L\) is a lattice of relative congruences if there are a quasivariety \(R\) of algebras and an algebra \(A\) such that \(L\) is isomorphic to \(\operatorname{Con}_RA\).
The author proves that every finite lattice is isomorphic to the lattice of relative congruences for some finite unar or Abelian group.

MSC:

06B05 Structure theory of lattices
08A30 Subalgebras, congruence relations
20K01 Finite abelian groups
08A60 Unary algebras
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