Principal congruences on semi-De Morgan algebras.(English)Zbl 0981.06007

A semi-De Morgan algebra is an algebra $$(L,\wedge, \vee, { }',0,1)$$ of type $$(2,2,1,0,0)$$ such that $$(L,\wedge,\vee,0,$$ $$1)$$ is a bounded distributive lattice and the following identities are satisfied: $0'=1,\quad 1'=0,\quad (x\vee y)'= x'\wedge y',\quad (x\wedge y)''= x''\wedge y'',\quad x'''= x'.$ The authors characterize those semi-De Morgan algebras which have only principal congruences (Theorem 3.14). In particular all such algebras are finite. The paper extends some of the results obtained by R. Beazer [Port. Math. 50, 75-86 (1993; Zbl 0801.06023)].

MSC:

 06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects) 08A30 Subalgebras, congruence relations 06D15 Pseudocomplemented lattices

Zbl 0801.06023
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