zbMATH — the first resource for mathematics

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 \(\text{Quord}(A)=\text{Con}(A)\).

06B15 Representation theory of lattices
06D05 Structure and representation theory of distributive lattices
08A55 Partial algebras
Full Text: EuDML
[1] BLOOM. S. L.: Varieties of ordered algebras. J. Comput. Sustem Sci. 13 (1976), 200-212. · Zbl 0337.06008
[2] CHAJDA I.-RACHUNEK J.: Relational characterizations of permutable and n-permutable varieties. Czechoslovak Math. J. 33 (1963), 505-508. · Zbl 0545.08011
[3] CHAJDA I.-PINUS A. G.: On quasiorders of universal algebras. Algebra i Logika 32 (1993), 308-325. · Zbl 0824.08002
[4] CZEDLl G.-LENKEHEGYI A.: On classes of ordered algebras and quasiorder distributivity. Acta Sci. Math. (Szeged) 46 (1983), 41-54. · Zbl 0541.06012
[5] JONSSON B.: On the representation of lattices. Math. Scand. 1 (1953), 193-206. · Zbl 0053.21304
[6] GRÄTZER G.-SGHMIDT E. T.: Characterizations of congruence lattices of abstract algebras. Acta Sci. Math. (Szeged) 24 (1963), 34-59. · Zbl 0117.26101
[7] GRÄTZER G.: General Lattice Theory. Akademie-Verlag, Berlin, 1978. · Zbl 0436.06001
[8] GRÄTZER G.: Universal Algebra. (2nd, Springer-Verlag, New York-Heidelberg-Berlin. 1979. · Zbl 0412.08001
[9] PUDLÁK P.-TUMA J.: Yeast graphs and fermentation of algebraic lattices. Latticce Theoru. Proc. Lattice Theoru Conf. (Szeged 1974). Colloq. Math. Soc. János Bolyai 14, North-Holland. Amsterdam, 1976, pp. 301-341.
[10] STONE M. H.: The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40 (1936), 35-111. · Zbl 0014.34002
[11] WHITMAN, PH. M.: Lattices, equivalence relations, and subgroups. Bull. Amer. Malh. Soc. 52 (1946), 507-522 · Zbl 0060.06505
[12] CZÉDLI G.: A Horn sentence for involution lattices of quasiorders. Order 11 (1994), 391-395. · Zbl 0817.06007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.