Congruence Lattice Research Articles

Congruences of finite semidistributive lattices

Congruences of finite semidistributive lattices

Subresiduated Nelson algebras

In this paper we generalize the well known relation between Heyting algebras and Nelson algebras in the framework of subresiduated lattices. In order to make it possible, we introduce the variety of subresiduated Nelson algebras. The main tool for its study is the construction provided by Vakarelov. Using it, we characterize the lattice of congruences of a subresiduated Nelson algebra through some of its implicative filters. We use this characterization to describe simple and subdirectly irreducible algebras, as well as principal congruences. Moreover, we prove that the variety of subresiduated Nelson algebras has equationally definable principal congruences and also the congruence extension property. Additionally, we present an equational base for the variety generated by the totally ordered subresiduated Nelson algebras. Finally, we show that there exists an equivalence between the algebraic category of subresiduated lattices and the algebraic category of centedred subresiduated Nelson algebras.

Properties of congruence lattices of graph inverse semigroups

From any directed graph [Formula: see text] one can construct the graph inverse semigroup [Formula: see text], whose elements, roughly speaking, correspond to paths in [Formula: see text]. Wang and Luo showed that the congruence lattice [Formula: see text] of [Formula: see text] is upper-semimodular for every graph [Formula: see text], but can fail to be lower-semimodular for some [Formula: see text]. We provide a simple characterization of the graphs [Formula: see text] for which [Formula: see text] is lower-semimodular. We also describe those [Formula: see text] such that [Formula: see text] is atomistic, and characterize the minimal generating sets for [Formula: see text] when [Formula: see text] is finite and simple.

Congruence filter pairs, equational filter pairs and adjoints

Abstract Filter pairs are a tool for creating and analyzing logics. A filter pair can be seen as a presentation of a logic, given by presenting its lattice of theories as the image of a lattice homomorphism, with certain properties ensuring that the resulting logic is substitution invariant. Every substitution invariant logic arises from a filter pair. Particular classes of logics can be characterized as arising from special classes of filter pairs. We consider so-called congruence filter pairs, i.e. filter pairs for which the domain of the lattice homomorphism is a lattice of congruences for some quasivariety. We show that the class of logics admitting a presentation by such a filter pair is exactly the class of logics having an algebraic semantics. We study the properties of a certain Galois connection coming with such filter pairs. We give criteria for a congruence filter pair to present a logic in some classes of the Leibniz hierarchy by means of this Galois connection, and its interplay with the Leibniz operator.

Linear elastic diatomic multilattices: Three-dimensional constitutive modeling and solutions of the shift vector equation

Diatomic multilattices are congruences of simple lattices each made out of atoms of two possible chemical species. We here constitutively characterize, in three dimensions, diatomic multilattices for the geometrically and materially linear elastic regime. We give the most generic expression for the energy for ( n + 1 ) diatomic multilattices and characterize explicitly the tensors present in such an expression for all 122 two-color point groups. For the specific case of diatomic 2- and 3-lattices, we delineate how one can solve the shift vector equation in the static case. For cases where the unique solution of the shift vector in terms of the strain tensor is not possible, we give conditions for the existence of solutions based on the standard Fredholm alternative theorem.

HIGHER MOMENT FORMULAE AND LIMITING DISTRIBUTIONS OF LATTICE POINTS

Abstract We establish higher moment formulae for Siegel transforms on the space of affine unimodular lattices as well as on certain congruence quotients of $\mathrm {SL}_d({\mathbb {R}})$ . As applications, we prove functional central limit theorems for lattice point counting for affine and congruence lattices using the method of moments.

Modal expansions of ririgs

Abstract In this paper, we introduce the variety of $I$-modal ririgs. We characterize the congruence lattice of its members by means of $I$-filters, and we provide a description of $I$-filter generation. We also provide an axiomatic presentation for the variety generated by chains of the subvariety of contractive $I$-modal ririgs. Finally, we introduce a Hilbert-style calculus for a logic with $I$-modal ririgs as an equivalent algebraic semantics and we prove that such a logic has the parametrized local deduction-detachment theorem.

Celebrating Loday’s associahedron

We survey Jean-Louis Loday’s vertex description of the associahedron, and its far reaching influence in combinatorics, discrete geometry, and algebra. We present in particular four topics where it plays a central role: lattice congruences of the weak order and their quotientopes, cluster algebras and their generalized associahedra, nested complexes and their nestohedra, and operads and the associahedron diagonal.

Multiplicative lattices, idempotent endomorphisms, and left skew braces

We show that, making use of multiplicative lattices and idempotent endomorphisms of an algebraic structure [Formula: see text], it is possible to derive several notions concerning [Formula: see text] in a natural way. The multiplicative lattice necessary here is the complete lattice of congruences of [Formula: see text] with multiplication given by commutator of congruences. Our main application is to the study of some notions concerning left skew braces.

On Freese’s technique

In this paper, we explore some applications of a certain technique (that we call Freese’ technique), which is a tool for identifying certain lattices as sublattices of the congruence lattice of a given algebra. In particular we will give sufficient conditions for two family of lattices (called the rods and the snakes) to be admissible as sublattices of a variety generated by a given algebra, extending an unpublished result of Freese and Lipparini.

Distributivity in congruence lattices of graph inverse semigroups

Let be a directed graph and be the graph inverse semigroup of . Luo and Wang (Semimodularity in congruence lattice of graph inverse semigroups, 2021) showed that the congruence lattice of any graph inverse semigroup is upper semimodular, but not lower semimodular in general. Anagnostopoulou-Merkouri, Mesyan and Mitchell (Properties of congruence lattices of graph inverse semigroups, 2022) characterized the directed graph for which is lower semimodular. In the present paper, we show that lower semimodularity, modularity and distributivity in the congruence lattice of any graph inverse semigroup are equivalent.

Measurable functions on σ-frames

We study (semi)measurable functions on σ-frames, extending the theory of real-valued functions from frames to σ-frames. The new objects of study are the σ-frame homomorphisms ▪ from the usual frame of reals into the congruence lattice of a σ-frame L, and its subclass of measurable functions L(R)→L. The desired extension faces two obstacles: (1) in general, σ-frames have no uncountable joins and are not pseudocomplemented; (2) σ-sublocales, that is, the subobjects in the dual category of the category of σ-frames and σ-frame homomorphisms, cannot be described as concrete subsets of the σ-frame L, unlike their counterpart in the category of locales, forcing us to work in the congruence lattice of L.Nevertheless, it is shown that, despite (1), the familiar method for generating real functions on frames via scales can be extended to arbitrary σ-frames. This is achieved by the new notions of σ-scale and finite σ-scale. It is also shown that, despite (2), the familiar insertion, extension and separation results for real-valued functions in several classes of frames (normal, extremally disconnected, G-perfect, F-perfect, perfectly normal) can be proved without involving uncountable joins and pseudocomplements, thus allowing their extension to measurable and semimeasurable functions.

Infinitely many new properties of the congruence lattices of slim semimodular lattices

Slim planar semimodular lattices (SPS lattices or slim semimodular lattices for short) were introduced by G. Grätzer and E. Knapp in 2007. More than four dozen papers have been devoted to these (necessarily finite) lattices and their congruence lattices since then. In addition to distributivity, the congruence lattices of SPS lattices satisfy seven known properties. Out of these seven properties, the first two were published by G. Grätzer in 2016 and 2020, the next four by the present author in 2021, and the seventh jointly by G. Grätzer and the present author in 2022. Here we give two infinite families of new properties of the congruence lattices of SPS lattices. These properties are independent. We also present stronger versions of these properties but not all of them are independent, and improve three out of the seven previously known properties. The approach is based on lamps, which we introduced in a 2021 paper. In addition to using the 2021 results, we need to prove some easy new lemmas on lamps.

Lattice theory of torsion classes: Beyond 𝜏-tilting theory

The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set t o r s A \mathsf {tors} A of torsion classes over a finite-dimensional algebra A A . We show that t o r s A \mathsf {tors} A is a complete lattice which enjoys very strong properties, as bialgebraicity and complete semidistributivity. Thus its Hasse quiver carries the important part of its structure, and we introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of t o r s A \mathsf {tors} A . In particular, we give a representation-theoretical interpretation of the so-called forcing order, and we prove that t o r s A \mathsf {tors} A is completely congruence uniform. When I I is a two-sided ideal of A A , t o r s ( A / I ) \mathsf {tors} (A/I) is a lattice quotient of t o r s A \mathsf {tors} A which is called an algebraic quotient, and the corresponding lattice congruence is called an algebraic congruence. The second part of this paper consists in studying algebraic congruences. We characterize the arrows of the Hasse quiver of t o r s A \mathsf {tors} A that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, we study in detail the case of preprojective algebras Π \Pi , for which t o r s Π \mathsf {tors} \Pi is the Weyl group endowed with the weak order. In particular, we give a new, more representation theoretical proof of the isomorphism between t o r s k Q \mathsf {tors} k Q and the Cambrian lattice when Q Q is a Dynkin quiver. We also prove that, in type A A , the algebraic quotients of t o r s Π \mathsf {tors} \Pi are exactly its Hasse-regular lattice quotients.

Congruence Lattices of Ideals in Categories and (Partial) Semigroups

This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley–Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories.

Extensions and Congruences of Partial Lattices

Abstract For a partial lattice L the so-called two-point extension is defined in order to extend L to a lattice. We are motivated by the fact that the one-point extension broadly used for partial algebras does not work in this case, i.e. the one-point extension of a partial lattice need not be a lattice. We describe these two-point extensions and prove several properties of them. We introduce the concept of a congruence on a partial lattice and show its relationship to the notion of a homomorphism and its connections with congruences on the corresponding two-point extension. In particular we prove that the quotient L/E of a partial lattice L by a congruence E on L is again a partial lattice and that the two-point extension of L/E is isomorphic to the quotient lattice of the two-point extension L * of L by the congruence on L * generated by E. Several illustrative examples are enclosed.

Construction of some algebras of logic by using fuzzy ideals in mv-modules

We study the lattice structure of fuzzy A-ideals in an mv-module M (fai (M), symbolically) and show that it is a complete Heyting lattice and so the set of its pseudocomplements forms a Boolean algebra. In the sequel, the properties of fuzzy congruences in an mv-module are investigated and using them some structural theorems are stated and proved. Finally, it is proved that fai (M) can be embedded into the lattice of fuzzy congruences.

On distributive lattices of simple ordered semirings

In this paper, we study principal ordered ideals of ordered semirings, and give their descriptions with some properties. The characterization of simple ordered semirings is studied, and a relation between a prime ordered ideal and filter has been developed. Finally, we characterize the ordered semirings which are distributive lattices of simple ordered semirings. Here, we find that the equivalence relations [Formula: see text] and [Formula: see text] induced, respectively, from principal filters and principal ordered ideals coincide as distributive lattice congruence in an ordered semiring [Formula: see text] whenever [Formula: see text] is a distributive lattice of simple ordered semirings.

Involutive symmetric Gödel spaces, their algebraic duals and logic

It is introduced a new algebra (A, otimes , oplus , *, rightharpoonup , 0, 1) called L_PG-algebra if (A, otimes , oplus , *, 0, 1) is L_P-algebra (i.e. an algebra from the variety generated by perfect MV-algebras) and (A,rightharpoonup , 0, 1) is a Gödel algebra (i.e. Heyting algebra satisfying the identity (x rightharpoonup y ) vee (y rightharpoonup x ) =1). The lattice of congruences of an L_PG -algebra (A, otimes , oplus , *, rightharpoonup , 0, 1) is isomorphic to the lattice of Skolem filters (i.e. special type of MV-filters) of the MV-algebra (A, otimes , oplus , *, 0, 1). The variety mathbf {L_PG} of L_PG -algebras is generated by the algebras (C, otimes , oplus , *, rightharpoonup , 0, 1) where (C, otimes , oplus , *, 0, 1) is Chang MV-algebra. Any L_PG -algebra is bi-Heyting algebra. The set of theorems of the logic L_PG is recursively enumerable. Moreover, we describe finitely generated free L_PG-algebras.

On the Uniqueness of Lattice Characterization of Groups

We analyze the problem of the uniqueness of characterization of groups by their weak congruence lattices. We discuss the possibility that the same algebraic lattice L acts as a weak congruence lattice of a group in more than one way, so that the corresponding diagonals are represented by different elements of L. If this is impossible, that is, if L can be interpreted as a weak congruence lattice of a group in a single way, we say that L is a sharp lattice. We prove that groups in many classes have a sharp weak congruence lattice. In particular, we analyze connections among isomorphisms of subgroup lattices of groups and isomorphisms of their weak congruence lattices. Summing up, we prove that there is a one-to-one correspondence between many known classes of groups and lattice-theoretic properties associated with each of these classes. Finally, an open problem is formulated related to the uniqueness of the element corresponding to the diagonal in the lattice of weak congruences of a group.