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This paper is devoted to a general investigation of congruences and ideals in effect algebras. One of our main results is the existence of an order isomorphism between Riesz congruences and Riesz ideals. We also answer an open question of Dvurečenskij and Pulmannová by showing that an ideal is a Riesz ideal if and only if it is closed under generalized Sasaki projections.
Let G be a plane bipartite graph and M(G) the set of perfect matchings of G. The Z-transformation graph of G is defined as a graph on M(G): M,M′∈M(G) are joined by an edge if and only if they differ only in one cycle that is the boundary of an inner face of G. A property that a certain orientation of the Z-transformation graph of G is acyclic implies a partially ordered relation on M(G). An equivalent...
This paper presents new proofs of two classic characterization theorems for families of ordered sets. The first is that any finite poset with no restriction isomorphic to $$\underline 2 + \underline 2 $$ has an interval representation. The second is that any finite poset with no restriction isomorphic to $$\underline 2 + \underline 2 $$ or to $$\underline 3 + \underline 1 $$ has...
We prove the NP-completeness of a weighted version of the jump number problem on two-dimensional orders, by reducing the Maximum Independent Set on cubic planar graphs, using a geometrical construction.
We study which infinite posets have simple cofinal subsets such as chains, or decompose canonically into such subsets. The posets of countable cofinality admitting such a decomposition are characterized by a forbidden substructure; the corresponding problem for uncountable cofinality remains open.
For a class C of finite lattices, the question arises whether any lattice in C can be embedded into some atomistic, biatomic lattice in C. We provide answers to the question above for C being, respectively, – the class of all finite lattices; – the class of all finite lower bounded lattices (solved by the first author's earlier work); – the class of all finite join-semidistributive lattices (this...
Let J be a fixed partially ordered set (poset). Among all posets in which J is join-dense and consists of all completely join-irreducible elements, there is an up to isomorphism unique greatest one, the Alexandroff completion L. Moreover, the class of all such posets has a canonical set of representatives, C0L, consisting of those sets between J and L which intersect each of the intervals Ij=[...
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