De Morgan algebras are universal. (English) Zbl 0618.06006
A concrete category K is called universal if the category of graphs and compatible mappings can be fully embedded into K. A bounded distributive lattice L with a unary operation \(\sim\) satisfying the identities \(\sim (a\wedge b)=\sim a\vee \sim b\), \(\sim (a\vee b)=\sim a\wedge \sim b\), \(\sim \sim a=a\) is called a de Morgan algebra. De Morgan algebras form a variety such that its lattice of subvarieties is a four-element chain [see J. A. Kalman; Trans. Am. Math. Soc. 87, 485-491 (1958; Zbl 0228.06003)]. The authors show that the variety of de Morgan algebras is universal. Moreover, the constructed embedding maps finite graphs to finite de Morgan algebras.
Reviewer: V.Koubek
MSC:
| 06D30 | De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects) |
| 18B15 | Embedding theorems, universal categories |
| 08B15 | Lattices of varieties |
Keywords:
universal concrete category; bounded distributive lattice; De Morgan algebras; lattice of subvarieties; variety of de Morgan algebrasCitations:
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