In this paper, after recalling the category {\bf PosAct}-$S$ of all poset acts over a pomonoid $S$; an $S$-act in Pos} posets, with action preserving monotone maps between them, some categorical properties are considered. particular, we describe limits and colimits such as products, coproducts, equalizers, coequalizers etc. category. Also, several kinds epimorphisms monomorphisms characterized ...

The quadratic Tits form, introduced by P. Gabriel [1] for quivers, Yu.A. Drozd [2] for posets, and M.M. Kleiner, A.V. Roiter [3] and Yu.A. Drozd [4] for a wide class of classification problems, plays an important role in representation theory. In particular, there are many results on connections between representation types of various objects and properties of the Tits forms. The reader interes...

‎in this paper some properties of weak regular injectivity for $s$-posets‎, ‎where $s$ is a pomonoid‎, ‎are studied‎. ‎the behaviour‎ ‎of different kinds of weak regular injectivity with products‎, ‎coproducts and direct sums is considered‎. ‎also‎, ‎some‎ ‎characterizations of pomonoids over which all $s$-posets are of‎ ‎some kind of weakly regular injective are obtained‎. ‎further‎, ‎we‎ ‎giv...

‎‎‎in this paper‎, ‎the‎ notion of injectivity with respect to order dense embeddings in ‎‎the category of $s$-posets‎, ‎posets with a monotone action of a‎ pomonoid $s$ on them‎, ‎is studied‎. ‎we give a criterion‎, ‎like the baer condition for injectivity of modules‎, ‎or skornjakov criterion for injectivity of $s$-sets‎, ‎for the order dense injectivity‎. ‎also‎, ‎we consider such injectivit...

S. Janson [Poset limits and exchangeable random posets, Combinatorica 31 (2011), 529–563] defined limits of finite posets in parallel to the emerging theory of limits of dense graphs. We prove that each poset limit can be represented as a kernel on the unit interval with the standard order, thus answering an open question of Janson. We provide two proofs: real-analytic and combinatorial. The co...

The weak order on the symmetric group naturally extends to a lattice on all integer binary relations. We first show that the subposet of this weak order induced by integer posets defines as well a lattice. We then study the subposets of this weak order induced by specific families of integer posets corresponding to the elements, the intervals, and the faces of the permutahedron, the associahedr...

In this paper we introduce a class of partially ordered sets, called differential posets, with many remarkable combinatorial and algebraic properties. (Terminology from lattice theory and the theory of partially ordered sets not explained here may be found in [St6].) The combinatorial properties of differential posets are concerned with the counting of saturated chains Xl < x2 < ... < x k or of...

The category of effect algebras is the Eilenberg-Moore category for the monad arising from the free-forgetful adjunction between categories of bounded posets and orthomodular posets. In the category of effect algebras, an observable is a morphism whose domain is a Boolean algebra. The characterization of subsets of ranges of observables is an open problem. For an interval effect algebra E, a wi...