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Distance function defined by Chang is an important tool for describing closeness and constructing topologies and uniformities on MV-algebras. Unfortunately, this function on residuated lattices is not good enough as on MV-algebras since it is not compatible with operations on residuated lattices. Based on this fact, the axioms of similarity operators and semi-norms are introduced on residuated lattices. By using the above two tools, uniformities and topologies are induced, respectively. Residuated lattices equipped with these uniformities (topologies) are proved to be uniform (topological) residuated lattices. Finally, two kinds of sequential completions for these uniformities are given and they are isomorphic.

We introduce a universal approach for applying the partition rank method, an extension of Tao’s slice rank polynomial method, to tensors that are not diagonal. This is accomplished by generalizing Naslund’s distinctness indicator to what we call a partition indicator. The advantages of partition indicators are two-fold: they diagonalize tensors that are constant when specified sets of variables are equal, and even in more general settings they can often substantially reduce the partition rank as compared to when a distinctness indicator is applied. The key to our discoveries is integrating the partition rank method with Möbius inversion on the lattice of partitions of a finite set. Through this we unify disparate applications of the partition rank method in the literature. We then use our theory to address a finite field analogue of a question of Erdős, thereby generalizing results of Hart and Iosevich and independently Shparlinski. Furthermore we generalize work of Pach et al. on bounding sizes of sets avoiding right triangles to bounding sizes of sets avoiding right k-configurations.

For every ordered set, we reconstruct the deck obtained by removal of the elements of rank r that are neither minimal nor maximal. Consequently, we also reconstruct the deck obtained by removal of the extremal, that is, minimal or maximal, elements. Finally, we reconstruct the ordered sets with a minmax pair of pseudo-similar points.

In this paper we provide a preliminary investigation of subclasses of bounded posets with antitone involution which are “pastings” of their maximal Kleene sub-lattices. Specifically, we introduce super-paraorthomodular lattices, namely paraothomodular lattices whose order determines, and it is fully determined by, the order of their maximal Kleene sub-algebras. It will turn out that the (spectral) paraorthomodular lattice of effects over a separable Hilbert space can be considered as a prominent example of such. Therefore, it arguably provides an algebraic/order theoretical rendering of complementarity phenomena between unsharp observables. A number of examples, properties and characterization theorems for structures we deal with will be outlined. For example, we prove a forbidden configuration theorem and we investigate the notion of commutativity for modular pseudo-Kleene lattices, examples of which are (spectral) paraorthomodular lattices of effects over finite-dimensional Hilbert spaces.

In order to provide a unified semantics for non-commutative algebraic logic, based on posets, Rump and Yang introduced the concept of quantum B-algebras. In this paper, we mainly consider the construction of quantum B-algebras over posets. We prove that a finite poset can support a quantum B-algebra if and only if its every connected component has a greatest element. However, such a result for infinite posets is not necessarily true. Under certain conditions, some interesting results for a poset to support quantum B-algebra are provided.

Effect algebras are certain ordered structures that serve as a general framework for studying the algebraic semantics of quantum logic. We study effect algebras which obtain suprema of countable monotone sequences – so-called (omega )-effect algebras. This assumption is necessary to capture basic (non-discrete) probabilistic concepts. We establish a free (omega )-completion of effect algebras (i.e., a left adjoint to the functor that forgets the existence of (omega )-suprema) and the existence of a tensor product in the category of (omega )-effect algebras. These results are obtained by means of so-called test spaces. Test spaces form a category that contains effect algebras as a reflective subcategory, but provides more space for constructions.

The cycle poset consists of the intervals of the cyclic permutation of the elements 1, 2, ..., n, ordered by inclusion. Suppose that F is a set of such intervals, none of them is a less than s others. The maximum size of F is determined under this condition. It is also shown that if the largest size of a set in this poset without containing a small subposet P is known, it solves the same problem, up to an additive constant, in the grid poset consisting of the pairs ((i,j) (1le i,jle n)) and ordered coordinate-wise.

It is known that the set of all nonnegative integers may be equipped with a total order that is chaotic in the sense that there is no monotone three-term arithmetic progressions. Such chaotic order must be so complicated that the resulting ordered set cannot be order isomorphic to the set of all nonnegative integers or the set of all integers with the standard order. In this paper, we completely characterize order structures of chaotic orders on the set of all nonnegative integers, as well as on the set of all integers and on the set of all rational numbers.

A barcode is a finite multiset of intervals on the real line. Jaramillo-Rodriguez (2023) previously defined a map from the space of barcodes with a fixed number of bars to a set of multipermutations, which presented new combinatorial invariants on the space of barcodes. A partial order can be defined on these multipermutations, resulting in a class of posets known as combinatorial barcode lattices. In this paper, we provide a number of equivalent definitions for the combinatorial barcode lattice, show that its Möbius function is a restriction of the Möbius function of the symmetric group under the weak Bruhat order, and show its ground set is the Jordan-Hölder set of a labeled poset. Furthermore, we obtain formulas for the number of join-irreducible elements, the rank-generating function, and the number of maximal chains of combinatorial barcode lattices. Lastly, we make connections between intervals in the combinatorial barcode lattice and certain classes of matchings.

A distributive lattice with zero is completely normal if its prime ideals form a root system under set inclusion. Every such lattice admits a binary operation ((x,y)mapsto xmathbin {smallsetminus }y) satisfying the rules (xle yvee (xmathbin {smallsetminus }y)) and ((xmathbin {smallsetminus }y)wedge (ymathbin {smallsetminus }x)=0) — in short a deviation. In this paper we study the following additional properties of deviations: monotone (i.e., isotone in x and antitone in y) and Cevian (i.e., (xmathbin {smallsetminus }zle (xmathbin {smallsetminus }y)vee (ymathbin {smallsetminus }z))). We relate those matters to finite separability as defined by Freese and Nation. We prove that every finitely separable completely normal lattice has a monotone deviation. We pay special attention to lattices of principal (ell )-ideals of Abelian (ell )-groups (which are always completely normal). We prove that for free Abelian (ell )-groups (and also free (Bbbk )-vector lattices) those lattices admit monotone Cevian deviations. On the other hand, we construct an Archimedean (ell )-group with strong unit, of cardinality (aleph _1), whose principal (ell )-ideal lattice does not have a monotone deviation.