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

Palma, Cândida; Santos, Raquel

Stud. Log. 67, No. 1, 75-88 (2001).

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)].

Reviewer: V.N.Salij (Saratov)

MSC:

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

Keywords:

pseudocomplementation; semi-De Morgan algebra; principal congruences

Citations:

Zbl 0801.06023