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