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Four notes on quasiorder lattices. (English) Zbl 0889.06001
The quasiorders, i.e., reflexive, transitive and compatible relations, of a (partial) algebra $$A$$ form a lattice $$\text{Quord}(A)$$ with an involution $$\rho \mapsto \rho^{-1}=\{\langle x,y\rangle \: \langle y,x\rangle \in \rho \}$$. It is shown that every algebraic lattice with involution is isomorphic to $$\text{Quord}(A)$$ for some partial algebra $$A$$. Any finite distributive lattice with involution is isomorphic to $$\text{Quord}(A)$$ for some finite algebra $$A$$ such that the quasiorders of $$A$$ are 3-permutable. Every distributive lattice with involution can be embedded in $$\text{Quord}(A)$$ for some set $$A$$. Any algebraic lattice is isomorphic to $$\text{Quord}(A)$$ for some algebra $$A$$ such that $$\text{Quord}(A)=\text{Con}(A)$$.

##### MSC:
 06B15 Representation theory of lattices 06D05 Structure and representation theory of distributive lattices 08A55 Partial algebras
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##### References:
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