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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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