For an arbitrary algebra A a new labelling, called the signed labelling, of the Hasse diagram of Con A is described. Under the signed labelling, each edge of the Hasse diagram of Con A receives a label from the set {+,−}. The signed labelling depends completely on a subset of the unary polynomials of A and its inspiration comes from semigroup theory. For finite algebras, the signed labelling co...

The Tamari lattice of order n can be defined by the set Dn of Dyck words endowed with the partial order relation induced by the well-known rotation transformation. In this paper, we study this rotation on the restricted set of Motzkin words. An upper semimodular join semilattice is obtained and a shortest path metric can be defined. We compute the corresponding distance between two Motzkin word...

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...

In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finite posets are shellable. A technique for labeling the edges of the Hasse diagram of certain lattices, due to...

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...