Previous |  Up |  Next


closure system; distributive lattice; lattices of $\Sigma-closed subsets; modular lattice; algebraic structures; $\Sigma$-closed subset; convex subset
Let $\Cal A =(A,F,R)$ be an algebraic structure of type $\tau$ and $\Sigma$ a set of open formulas of the first order language $L(\tau)$. The set $C_\Sigma(\Cal A)$ of all subsets of $A$ closed under $\Sigma$ forms the so called lattice of $\Sigma$-closed subsets of $\Cal A$. We prove various sufficient conditions under which the lattice $C_\Sigma(\Cal A)$ is modular or distributive.
[1] Chajda I.: A note on varieties with distributive subalgebra lattices. Acta Univ. Palack. Olomouc, Fac. Rer. Natur., Matematica 31 (1992), 25-28. MR 1212602 | Zbl 0777.08001
[2] Chajda I., Emanovský P.: $\Sigma$-isomorphic algebraic structures. Mathem. Bohemica 120 (1995), 71-81. MR 1336947 | Zbl 0833.08001
[3] Emanovský P.: Convex isomorphic ordered sets. Mathem. Bohemica 118 (1993), 29-35. MR 1213830
[4] Evans T., Ganter B.: Varieties with modular subalgebra lattices. Bull. Austral. Math. Soc. 28 (1993), 247-254. DOI 10.1017/S0004972700020918 | MR 0729011
[5] Jakubíková-Studenovská D.: Convex subsets of partial monounary algebras. Czech. Math. J. 38 (1988), no. 113, 655-672. MR 0962909
[6] Marmazajev V.I.: The lattice of convex sublattices of a lattice. Mezvužovskij naučnyj sbornik 6. Saratov, 1986, pp. 50-58. (In Russian.) MR 0957970
Partner of
EuDML logo