A Horn sentence for involution lattices of quasiorders. (English) Zbl 0817.06007

Czédli, Gábor

Order 11, No. 4, 391-395 (1994).

A triplet \({\mathcal L}= (L, \leq, ^{-1})\) is called an involution lattice if \((L, \leq)\) is a lattice and \(^{-1}\) is a lattice automorphism satisfying the identity \((x^{-1})^{-1}= x\). A quasiorder of \(A\) is a reflexive and transitive binary relation on a set \(A\). The quasiorders of a set \(A\) form a lattice \(\text{Quord }A\) with an involution \(\rho\to \rho^{-1}= \{\langle x, y\rangle; \langle y, x\rangle\in \rho\}\). The author gives an explicit Horn sentence which holds in \(\text{Quord }A\) for any set \(A\) but does not hold in all involution lattices. Hence, not every involution lattice can be embedded in \(\text{Quord }A\) for some set \(A\).

Reviewer: I.Chajda (Přerov)

MSC:

06B15	Representation theory of lattices
06B25	Free lattices, projective lattices, word problems

Keywords:

embedding; involution lattice; quasiorder; Horn sentence

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