We introduce a family of posets which generate Lie poset subalgebras $$A_{n-1}=\mathfrak {sl}(n)$$ whose index can be realized topologically. In particular, if $$\mathcal {P}$$ is such toral poset, then it has simplicial realization homotopic to wedge sum d one-spheres, where the corresponding type-A algebra $$\mathfrak {g}_A(\mathcal {P})$$ . Moreover, when Frobenius, its spectrum binary, that...

For each word w in the Fibonacci lattices Fib(r) and Z (r) we partition the interval ^ 0; w] in Fib(r) into subposets called r-Boolean posets. In the case r = 1 those subposets are isomorphic to Boolean algebras. We also partition the interval ^ 0; w] in Z (r) into certain spanning trees of the r-Boolean posets. A bijection between those intervals is given in which each r-Boolean poset in Fib(r...

We show that there are n! matchings on 2n points without socalled left (neighbor) nestings. We also define a set of naturally labeled (2+2)free posets and show that there are n! such posets on n elements. Our work was inspired by Bousquet-Mélou, Claesson, Dukes and Kitaev [J. Combin. Theory Ser. A. 117 (2010) 884–909]. They gave bijections between four classes of combinatorial objects: matching...

We study observables as r-D-homomorphisms de®ned on Boolean D-posets of subsets into a Boolean Dposet. We show that given an atomic r-complete Boolean D-poset P with the countable set of atoms there exist a r-complete Boolean D-poset of subsets S and a r-D-homomorphism h from S onto P, more precisely we can choose S ˆ B…R†, which gives an analogy of the Loomis± Sikorski representation theorem f...

All the posets/lattices considered here are finite with element 0. An element x of a poset satisfying certain properties is deletable if P − x is a poset satisfying the same properties. A class of posets is reducible if each poset of this class admits at least one deletable element. When restricted to lattices, a class of lattices is reducible if and only if one can go from any lattice in this ...

The interval number i(P) of a poset P is the smallest t such that P is a containment of sets that are unions of at most t real intervals. For the special poset Bn(k) consisting of the singletons and ksubsets of an n-element set, ordered by inclusion, i(Bn(k)) = min {k, n − k + 1} if |n/2 − k | ≥ n/2 − (n/2). For bipartite posets with n elements or n minimal elements, i(P) ≤  n lgn − lglgn  + ...

From the point of view of modal logic, coalgebraic logic over posets is the natural coalgebraic generalisation of positive modal logic. From the point of view of coalgebra, posets arise if one is interested in simulations as opposed to bisimulations. From a categorical point of view, one moves from ordinary categories to enriched categories. We show that the basic setup of coalgebraic logic ext...

As a generalization of countably C-approximating posets, the concept of countably QC-approximating posets is introduced. With the countably QC-approximating property, some characterizations of generalized completely distributive lattices and generalized countably approximating posets are given. The main results are as follows: (1) a complete lattice is generalized completely distributive if and...

The method of information systems is extended from algebraic posets to continuous posets by taking a set of tokens with an ordering that is transitive and interpolative but not necessarily reflexive. This develops results of Raney on completely distributive lattices and of Hoofman on continuous Scott domains, and also generalizes Smyth’s “R-structures”. Various constructions on continuous poset...

It is a well known fact that a supersolvable lattice is ELoshellable. Hence a supersolvable lattice (resp., its Stanley-Reisner ring) is Cohen-Macaulay. We prove that if L is a supersolvable lattice such that all intervals have non-vanishing Mt~bius number, then for an arbitrary element x e L the poser L {x} is also Cohen-Macaulay. Posets with this property are called 2-Cohen-Macaulay posets. I...