Let L be a slim, planar, semimodular lattice (slim means that it does not contain an $${{\textsf{M}}}_3$$ -sublattice). We call the interval $$I = [o, i]$$ of rectangular, if there are complementary $$a, b \in I$$ such is to left b. claim rectangular slim also lattice. will present some applications, including recent result G. Czédli. In paper with E. Knapp about dozen years ago, we introduced ...

We discuss the relationship between the notion of independence as defined in lattice theory, matroid theory, and Boolean algebras. Collections of Boolean sub-algebras {At} are indeed endowed with the following independence relation (IB) ∩At 6= ∧ with ∧ the initial element of the Boolean algebra they belong to. However, those collections can be given several algebraic interpretations in terms of...

We define a new object, called a signed poset, that bears the same relation to the hyperoctahedral group B n (i.e., signed permutations on n letters), as do posets to the symmetric group S n. We then prove hyperoctahedral analogues of the following results: (1) the generating function results from the theory of P-partitions; (2) the fundamental theorem of finite distributive lattices (or Birkho...

In 2010, Gábor Czédli and E. Tamás Schmidt mentioned that the best cover-preserving embedding of a given semimodular lattice is not known yet [A lattices into geometric lattices. Advances in Mathematics, 225, 2455–2463 (2010)]. That to say: What are G such finite L has with smallest ∣G∣? this paper, we propose an algorithm calculate all extending prove length number atoms every equal non-zero j...

The question of which semigroups have lower semimodular lattice of subsemigroups has been open since the early 1960’s, when the corresponding question was answered for modularity and for upper semimodularity. We provide a characterization of such semigroups in the language of principal factors. Since it is easily seen (and has long been known) that semigroups for which Green’s relation J is tri...

A matroid has been one of the most important combinatorial structures since it was introduced by Whitney as an abstraction linear independence. As property a matroid, can be characterized several different (but equivalent) axioms, such augmentation, base exchange, or rank axiom. supermatroid is generalization defined on lattices. Here, central question whether equivalent axioms similar to matro...

In the early forties, R. P. Dilworth proved his famous result: Every finite distributive lattice D can be represented as the congruence lattice of a finite lattice L. In one of our early papers, we presented the first published proof of this result; in fact we proved: Every finite distributive lattice D can be represented as the congruence lattice of a finite sectionally complemented lattice L....

A planar (upper) semimodular lattice L is slim if the five-element nondistributive modular M3 does not occur among its sublattices. (Planar lattices are finite by definition.) Slim rectangular as particular were defined G. Grätzer and E. Knapp in 2007. In 2009, they also proved that congruence of with at least three elements same those lattices. order to provide an effective tool for studying t...

Let L be a finite geometric lattice of rank n with rank function r. (For definitions, see e.g., [3, Chapter 2], [4], or [1, Chapter 4].) An element x s L is called a modular element if it forms a modular pair with every y e L , i.e., if a<~y then a V ( x A y ) = (a v x )Ay . Recall that in an upper semimodular lattice (and thus in a geometric lattice) the relation of being a modular pair is sym...

We study abstract properties of intervals in the complete lattice of all meet-closed subsets ( -subsemilattices) of a -(meet-)semilattice S, where is an arbitrary cardinal number. Any interval of that kind is an extremally detachable closure system (that is, for each closed set A and each point x outside A, deleting x from the closed set generated by A and x leaves a closed set). Such closure s...