arXiv subjects

Gabor Czedli

Publications and source records attributed to Gabor Czedli.

5 recordsLinked to original sources

arXiv2015

Cometic functors for small concrete categories and an application

Our goal is to derive some families of maps, also known as functions, from injective maps and surjective maps; this can be useful in various fields of mathematics. Let A be a small concrete category. We define a functor F, cometic functor, from A to the category Set and a natural transformation π, called cometic projection, from F to the inclusion functor of A into Set such that the F-image of every monomorphism A is an injective map and the components of πare surjective maps. Also, we give a nontrivial application of F and π.

math.CT↗

arXiv2013

CD-independent subsets in meet-distributive lattices

A subset $X$ of a finite lattice $L$ is CD-independent if the meet of any two incomparable elements of $X$ equals 0. In 2009, Czédli, Hartmann and Schmidt proved that any two maximal CD-independent subsets of a finite distributive lattice have the same number of elements. In this paper, we prove that if $L$ is a finite meet-distributive lattice, then the size of every CD-independent subset of $L$ is at most the number of atoms of $L$ plus the length of $L$. If, in addition, there is no three-element antichain of meet-irreducible elements, then we give a recursive description of maximal CD-independent subsets. Finally, to give an application of CD-independent subsets, we give a new approach to count islands on a rectangular board.

math.RA↗

arXiv2013

The ordered set of principal congruences of a countable lattice

For a lattice L, let Princ L denote the ordered set of principal congruences of L. In a pioneering paper, G. Gratzer characterized the ordered sets Princ L of finite lattices L; here we do the same for countable lattices. He also showed that each bounded ordered set H is isomorphic to Princ L of a bounded lattice L. We prove a related statement: if an ordered set H with least element is the union of a chain of principal order ideals, then H is isomorphic to Princ L of some lattice L.

math.RA↗

arXiv2013

Composition series in groups and the structure of slim semimodular lattices

Let H and K be finite composition series of a group G. The intersections H_i\cap K_j of their members form a lattice CSL(H,\K) under set inclusion. Improving the Jordan-Hölder theorem, G. Grätzer, J.B. Nation and the present authors have recently shown that H and K determine a unique permutation pi such that, for all i, the i-th factor of H is "down-and-up projective" to the pi(i)-th factor of K. Equivalent definitions of pi were earlier given by R.P. Stanley and H. Abels. We prove that pi determines the lattice CLS(H,K). More generally, we describe slim semimodular lattices, up to isomorphism, by permutations, up to an equivalence relation called "sectionally inverted or equal". As a consequence, we prove that the abstract class of all CSL(H,K) coincides with the class of duals of all slim semimodular lattices.

math.RA↗

arXiv2012

Varieties whose tolerances are homomorphic images of their congruences

The homomorphic image of a congruence is always a tolerance (relation) but, within a given variety, a tolerance is not necessarily obtained this way. By a Maltsev-like condition, we characterize varieties whose tolerances are homomorphic images of their congruences (TImC). As corollaries, we prove that the variety of semilattices, all varieties of lattices, and all varieties of unary algebras have TImC. We show that a congruence n-permutable variety has TImC if and only if it is congruence permutable, and construct an idempotent variety with a majority term that fails TImC.

math.RA↗