Abstract: A NaP-preference (necessary and possible preference) on a set A is a pair ${\left(\succsim^{^{_N}}\!,\,\succsim^{^{_P}}\!\right)}$ of binary relations on A such that its necessary component ${\succsim^{^{_N}} \!\!}$ is a partial preorder, its possible component ${\succsim^{^{_P}} \!\!}$ is a completion of ${\succsim^{^{_N}} \!\!}$ , and the two components jointly satisfy natural forms of mixed completeness and mixed transitivity. We study additional mixed transitivity properties of a NaP-preference ${\left(\succsim^{^{_N}}\!,\,\succsim^{^{_P}}\!\right)}$ , which culminate in the full transitivity of its possible component ${\succsim^{^{_P}} \!\!}$ . Interval orders and semiorders are strictly related to these properties, since they are the possible components of suitably transitive NaP-preferences. Further, we introduce strong versions of interval orders and semiorders, which are characterized by enhanced forms of mixed transitivity, and use a geometric approach to compare them to other well known preference relations.

Abstract: A quasi-orthomodular poset is defined to be a poset with 0 equipped with an orthogonality relation satisfying certain axioms. The goal of the paper is to compare such posets (also semilattices, nearlattices and lattices) with several other kinds of posets having an appropriate structure and already known in the literature: generalized orthomodular posets and lattices, generalized orthoalgebras, sectionally orthocomplemented, sectionally orthomodular and relatively orthocomplemented posets and meet semilattices, semi-orthomodular lattices, weak BCK-algebras.

Abstract: Canonical extensions of (bounded) lattices have been extensively studied, and the basic existence and uniqueness theorems for these have been extended to general posets. This paper focuses on the intermediate class ${{\boldsymbol{\mathcal{S}}}}_{\wedge}$ of (unital) meet semilattices. Any ${\mathbf S}\in {{\boldsymbol{\mathcal{S}}}}_{\wedge}$ embeds into the algebraic closure system Filt(Filt(S)). This iterated filter completion, denoted Filt2(S), is a compact and ${\textstyle{\bigvee}\,}{\textstyle{\bigwedge}\,}$ -dense extension of S. The complete meet-subsemilattice S δ of Filt2(S) consisting of those elements which satisfy the condition of ${\textstyle{\bigwedge}\,}{\textstyle{\bigvee}\,}$ -density is shown to provide a realisation of the canonical extension of S. The easy validation of the construction is independent of the theory of Galois connections. Canonical extensions of bounded lattices are brought within this framework by considering semilattice reducts. Any S in ${{\boldsymbol{\mathcal{S}}}}_{\wedge}$ has a profinite completion, ${\rm Pro}_{{{\boldsymbol{\mathcal{S}}}}_{\wedge}}({\mathbf S})$ . Via the duality theory available for semilattices, ${\rm Pro}_{{{\boldsymbol{\mathcal{S}}}}_{\wedge}}({\mathbf S})$ can be identified with Filt2(S), or, if an abstract approach is adopted, with ${\mathbb F_{\sqcup}}({\mathbb F_{\sqcap}}({\mathbf S}))$ , the free join completion of the free meet completion of S. Lifting of semilattice morphisms can be considered in any of these settings. This leads, inter alia, to a very transparent proof that a homomorphism between bounded lattices lifts to a complete lattice homomorphism between the canonical extensions. Finally, we demonstrate, with examples, that the profinite completion of S, for ${\mathbf S} \in {{\boldsymbol{\mathcal{S}}}}_{\wedge}$ , need not be a canonical extension. This contrasts with the situation for the variety of bounded distributive lattices, within which profinite completion and canonical extension coincide.

Abstract: This paper is devoted to the semilattice ordered \(\mathcal{V}\)-algebras of the form (A, Ω, + ), where + is a join-semilattice operation and (A, Ω) is an algebra from some given variety \(\mathcal{V}\). We characterize the free semilattice ordered algebras using the concept of extended power algebras. Next we apply the result to describe the lattice of subvarieties of the variety of semilattice ordered \(\mathcal{V}\)-algebras in relation to the lattice of subvarieties of the variety \(\mathcal{V}\).

Abstract: The N cardinality k ideals of any w-element poset (k ≤ w fixed) can be enumerated in time O(Nw 3). The corresponding bound for k-element subtrees of a w-element tree is O(Nw 5). An algorithm is described that by the use of wildcards displays all order ideals of a poset in a compact manner, i.e. not one by one.

Abstract: We give a shorter proof of the fact, that Bergman complexes of matroids can be subdivided to realizations of the nested set complexes of the lattice of flats Then, we present a direct sum decomposition into connected summands of the matroid types of faces of Bergman complexes

Abstract: We prove two theorems concerning incidence posets of graphs, cover graphs of posets and a related graph parameter. First, answering a question of Haxell, we show that the chromatic number of a graph is not bounded in terms of the dimension of its incidence poset, provided the dimension is at least four. Second, answering a question of Křiž and Nesetřil, we show that there are graphs with large girth and large chromatic number among the class of graphs having eye parameter at most two.

Abstract: A theorem of Sands, Sauer, and Woodrow, extending the Gale–Shapley theorem, states that if G is a digraph whose arc set is the union of the arc sets of two posets, then G has a kernel. We prove a weighted version of this theorem.

Abstract: If $ {\mathbb {V}} $ is a subvariety of $ {\mathbb {BL}} $ generated by a class of standard BL-algebras, then $ {\mathbb {V}} $ is generated by a finite class of standard BL-algebras.

Abstract: One of central issues in extremal set theory is Sperner’s theorem and its generalizations. Among such generalizations is the best-known LYM (also known as BLYM) inequality and the Ahlswede–Zhang (AZ) identity which surprisingly generalizes the BLYM into an identity. Sperner’s theorem and the BLYM inequality has been also generalized to a wide class of posets. Another direction in this research was the study of more part Sperner systems. In this paper we derive AZ type identities for regular posets. We also characterize all maximum 2-part Sperner systems for a wide class of product posets.

Abstract: Inspired by engineering of high-speed switching with quality of service, this paper introduces a new approach to classify finite lattices by the concept of cut-through coding. An n-ary cut-through code of a finite lattice encodes all lattice elements by distinct n-ary strings of a uniform length such that for all j, the initial j encoding symbols of any two elements x and y determine the initial j encoding symbols of the meet and join of x and y. In terms of lattice congruences, some basic criteria are derived to characterize the n-ary cut-through codability of a finite lattice. N-ary cut-through codability also gives rise to a new classification of lattice varieties and in particular, defines a chain of ideals in the lattice of lattice varieties.

Abstract: This paper builds on work characterising digraphs with rich transitivity properties. When such digraphs are equipped with a rank function onto a colexicographic power of ℤ they arise as direct limits of better understood structures. These structures, it will be seen, contain sufficient information to describe the cycles of the full structure.

Abstract: For an n ×n matrix algebra over a totally ordered integral domain, necessary and sufficient conditions are derived such that the entrywise lattice order on it is the only lattice order (up to an isomorphism) to make it into a lattice-ordered algebra in which the identity matrix is positive. The conditions are then applied to particular integral domains. In the second part of the paper we consider n ×n matrix rings containing a positive n-cycle over totally ordered rings. Finally a characterization of lattice-ordered matrix ring with the entrywise lattice order is given.

Abstract: We prove a conjecture of Reinhold: that a finite lattice is isomorphic to an interval in the lattice of topologies on some set if and only if it is isomorphic to an interval in the lattice of topologies on a finite set.

Abstract: An involutive residuated lattice (IRL) is a lattice-ordered monoid possessing residual operations and a dualizing element. We show that a large class of self-dual lattices may be endowed with an IRL structure, and give examples of lattices which fail to admit IRLs with natural algebraic conditions. A classification of all IRLs based on the modular lattices M n is provided.

Abstract: The linear discrepancy of a poset P, denoted ld(P), is the minimum, over all linear extensions L, of the maximum distance in L between two elements incomparable in P. With r denoting the maximum vertex degree in the incomparability graph of P, we prove that \({\rm ld}(P)\le \left\lfloor (3r-1)/2 \right\rfloor\) when P has width 2. Tanenbaum, Trenk, and Fishburn asked whether this upper bound holds for all posets. We give a negative answer using a randomized construction of bipartite posets whose linear discrepancy is asymptotic to the trivial upper bound 2r − 1. For products of chains, we give alternative proofs of results obtained independently elsewhere.

Abstract: It has been shown that the h, k-equal partition lattice \(\tilde \Pi_n^{h, k}\) is EL-shellable when h < k. We produce an EL-shelling for \(\tilde \Pi_n^{h, k}\) when n ≥ h ≥ k ≥ 2 and observe that, in this shelling, there are no weakly decreasing chains. This shows that \(\tilde \Pi_n^{h, k}\) is contractible for such values of h and k, which can also be seen by the fact that \(\tilde \Pi_n^{h, k}\) is noncomplemented.

Abstract: In this paper we give an algorithm to determine, for any given suborder closed class of series parallel posets, a structure theorem for the class. We refer to these structure theorems as structural descriptions. This work builds on work of Robertson, Seymour, Thomas, and especially Nigussie on trees. Stated differently, this paper gives an analogue of Nigussie’s Tree Algorithm for series parallel posets.

Abstract: This paper proves that, if there is an ordered set P that is not reconstructible, then each of its two-point-deleted subsets must be isomorphic to another one, or, it must violate a condition that is related to, but weaker than, rigidity. The conditions are inspired by an argument by Bollobas and they provide a connection between positive reconstruction results and partial counterexamples that was, so far, nonexistent in order reconstruction.

Abstract: Compact regular frames are always spatial. In this note we present a method for constructing non-spatial frames. As an application we show that there is a countably compact (and hence pseudocompact) completely regular frame which is not spatial.

Abstract: We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the standard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. Towards obtaining the topological duality, we develop a universal construction which associates to an arbitrary lattice two distributive lattice envelopes with a Galois connection between them. This is a modification of a construction of the injective hull of a semilattice by Bruns and Lakser, adjusting their concept of 'admissibility' to the finitary case. Finally, we show that the dual spaces of the distributive envelopes of a lattice coincide with completions of quasi-uniform spaces naturally associated with the lattice, thus giving a precise spatial meaning to the distributive envelopes.

Abstract: We discuss some properties of a subposet of the Tamari lattice introduced by Pallo (1986), which we call the comb poset. We show that three binary functions that are not well-behaved in the Tamari lattice are remarkably well-behaved within an interval of the comb poset: rotation distance, meets and joins, and the common parse words function for a pair of trees. We relate this poset to a partial order on the symmetric group studied by Edelman (1989). R´ esumNous discutons d'un subposet du treillis de Tamari introduit par Pallo. Nous appellons ce poset le comb poset. Nous montrons que trois fonctions binaires qui ne se comptent pas bien dans le trellis de Tamari se comptent bien dans un intervalle du comb poset : distance dans le trellis de Tamari, le supremum et l'infimum et les parsewords communs. De plus, nous discutons un rapport entre ce poset et un ordre partiel dans le groupe sym´ etrique ´´ par Edelman.

Abstract: We generalize the $\frac{1}{3}$ – $\frac{2}{3}$ conjecture from partially ordered sets to antimatroids: we conjecture that any antimatroid has a pair of elements x,y such that x has probability between $\frac{1}{3}$ and $\frac{2}{3}$ of appearing earlier than y in a uniformly random basic word of the antimatroid. We prove the conjecture for antimatroids of convex dimension two (the antimatroid-theoretic analogue of partial orders of width two), for antimatroids of height two, for antimatroids with an independent element, and for the perfect elimination antimatroids and node search antimatroids of several classes of graphs. A computer search shows that the conjecture is true for all antimatroids with at most six elements.

Abstract: We provide a complete classification of solvable instances of the equational unification problem over De Morgan and Kleene algebras with respect to unification type. The key tool is a combinatorial characterization of finitely generated projective De Morgan and Kleene algebras.

Abstract: We explore a question related to the celebrated Erdős-Szekeres Theorem and develop a geometric approach to answer it. Our main object of study is the Erdős-Szekeres Tableau, or EST, of a number sequence. An EST is the sequence of integral points whose coordinates record the length of the longest increasing and longest decreasing subsequence ending at each element of the sequence. We define the Order Poset of an EST in order to answer the question: What information about the sequence can be recovered by its EST?

Abstract: Four distinct elements a, b, c, and d of a poset form a diamond if \(a< b