Article

Full entry | PDF   (0.6 MB)
Keywords:
closure system; isomorphism; lattice of $\Sigma$-closed subsets; subalgebras; ideals; algebraic structure; $\Sigma$-closed subset; $\Sigma$-isomorphic structures
Summary:
For an algebraic structure $\A=(A,F,R)$ or type $\t$ and a set $\Sigma$ of open formulas of the first order language $L(\t)$ we introduce the concept of $\Sigma$-closed subsets of $\A$. The set $\C_\Sigma(\A)$ of all $\Sigma$-closed subsets forms a complete lattice. Algebraic structures $\A$, $\B$ of type $\t$ are called $\Sigma$-isomorphic if $\C_\Sigma(\A)\cong\C_\Sigma(\B)$. Examples of such $\Sigma$-closed subsets are e.g. subalgebras of an algebra, ideals of a ring, ideals of a lattice, convex subsets of an ordered or quasiordered set etc. We study $\Sigma$-isomorphic algebraic structures in dependence on the properties of $\Sigma$.
References:
[1] Birkhoff G., Bennet M. K.: The convexity lattice of a poset. Order 2 (1985), 223-242. MR 0824696
[2] Blum E. K., Estes D. R.: A generalization of the homomorphism concepts. Algebra Univ. 7 (1977), 143-161. DOI 10.1007/BF02485424 | MR 0434926
[3] Chajda I.: Lattices in quasiordered sets. Acta Univ. Palack. Olom. 31 (1992), 6-12. MR 1212600 | Zbl 0773.06002
[4] Emanovský P.: Convex isomorphic ordered sets. Matem. Bohem. 118 (1993), 29-35. MR 1213830
[5] Emanovský P.: Convex isomorphism of q-lattices. Matem. Bohem. 118 (1993), 37-42. MR 1213831 | Zbl 0780.06002
[6] Grätzer G.: Universal algebra. (2nd edition), Springer-Verlag, 1979. MR 0538623
[7] Jakubíková-Studenovská D.: Convex subsets of partial monounary algebras. Czech. Math. J. 38 (113) (1988), 655-672. MR 0962909
[8] Maľcev A. I: Algebraic systems. Nauka, Moskva, 1970. (In Russian.) MR 0282908
[9] Marmazajev V. I.: The lattice of convex sublattices of a lattice. Mezvužovskij naučnyj sbornik 6. Saratov, 1986, pp. 50-58. (In Russian.) MR 0957970
[10] Snášel V.: $\lambda$-lattices. PhD - thesis. Palacky University, Olomouc, 1991.
[11] Chajda I., Halas R.: Genomorphism of lattices and semilattices. Acta-UPO. To appear. MR 1385742

Partner of