For a positive integer \(n\), let SCL\((n)=\{|\)Con \((L)|: L\) is an \(n\)-element lattice\(\}\) stand for the set of Sizes of the Congruence Lattices of \(n\)-element lattices. The \(k\)-th Largest Number of Congruences of \(n\)-element lattices, denoted by lnc \((n, k)\), is the \(k\)-th largest member of SCL \( (n)\). Let \((n_1,\dots,n_6):=(1,4,5,6,6,7)\), and let \(n_k:=k\) for \(k\geq 7\). In 1997, R. Freese proved that for \(n\geq n_1=1\), lnc \( (n, 1)=2^{n-1}\). For \(n\geq n_2\), the present author gave lnc \((n, 2)\). For \(k=3,4,5\) and \(n\geq n_k\), C. Mureşan and J. Kulin determined lnc \((n, k)\) in their 2020 paper. For \(k\leq 5\) and \(n\geq n_k\), the above-mentioned authors described the \(n\)-element lattices witnessing lnc \((n, k)\), too. For all positive integers \(k\) and \(n \geq n_k\), this paper determines lnc \((n, k)\) and presents the lattices that witness it. It turns out that, for each fixed \(k\), the quotient lcd \((k):=\) lnc \((n, k)/\) lnc \((n, 1)\) does not depend on \(n\geq n_k\). Furthermore, lcd \((k)\) converges to \(1/8\) as \(k\) tends to infinity.

We show that the family of systoles of hyperbolic surfaces associated with congruence lattices in \operatorname{SL}_{2}(\mathbb{Z}) have asymptotically minimal crossing number.

In previous work we have studied minimal prime spectra, as well as extensions of universal algebras whose term condition commutator behaves like the modular commutator in the sense that it is commutative and distributive with respect to arbitrary joins, while modularity does not even need to be enforced on their congruence lattices, let alone on those of the members of the variety they generate. Commutator lattices, defined by Czelakowski in 2008, are commutative multiplicative lattices having as prototype the algebraic structure of the congruence lattice of a such an algebra. Considering the prime elements with respect to the commutator operation, we obtain algebraic characterizations for minimal primes, then study the Stone and flat topologies on the set of minimal primes in a commutator lattice. We also prove abstract versions of congruence extension properties, actually of the general case of arbitrary morphisms instead of algebra embeddings, by means of complete join–semilattice morphisms between commutator lattices. We thus obtain abstractions for our results on congruence lattices and generalizations for results on frames and quantales, but also further cases in which these results hold. Furthermore, we investigate the lattice structures of these topologies as sublattices of the power sets of the sets of (minimal) primes.

Not every finite distributive lattice is isomorphic to the congruence lattice of a finite semidistributive lattice. This note provides a construction showing that many of these finite distributive lattices are isomorphic to congruence lattices of infinite semidistributive lattices.

In this paper we investigate two logics (and their fragments) from an algebraic point of view. The two logics are: MALL (multiplicative-additive Linear Logic) and LL (classical Linear Logic). Both logics turn out to be strongly algebraizable in the sense of Blok and Pigozzi and their equivalent algebraic semantics are, respectively, the variety of Girard algebras and the variety of girales. We show that any variety of girales has a TD-term and hence equationally definable principal congruences. Also we investigate the structure of the algebras in question, thus obtaining a representation theorem for Girard algebras and girales. We also prove that congruence lattices of girales are really congruence lattices of Heyting algebras, thus determining the simple and subdirectly irreducible girales. Finally we introduce a class of examples showing that the variety of girales contains infinitely many nonisomorphic finite simple algebras.

If M is an algebra in a semidegenerate congruence-modular variety V, then the set Con(M) of congruences of M is an integral complete l-groupoid (= icl-groupoid). For any morphism f:M→N of V, consider the map f•:Con(M)→Con(N), where, for each congruence ε of M, f•(ε) is the congruence of N generated by f(ε). Then, f• is a semimorphism of icl-groupoids, i.e., it preserves the arbitrary joins and the top congruences. The neo-commutative icl-groupoids were introduced recently by the author as an abstraction of the lattices of congruences of Kaplansky neo-commutative rings. In this paper, we define the admissible semimorphisms of icl-groupoids. The basic construction of the paper is a covariant functor defined by the following: (1) to each semiprime and neo-commutative icl-groupoid A, we assign a coherent frame R(A) of radical elements of A; and (2) to an admissible semimorphism of icl-groupoids u:A→B, we assign a coherent frame morphism uρ:R(A)→R(B). By means of this functor, we transfer a significant amount of results from coherent frames and coherent frame morphisms to the neo-commutative icl-groupoids and their admissible semimorphisms. We study the m-prime spectra of neo-commutative icl-groupoids and the going-down property of admissible semimorphisms. Using some transfer properties, we characterize some classes of admissible semimorphisms of icl-groupoids: Baer and weak-Baer semimorphisms, quasi r-semimorphisms, quasi r*-semimorphisms, quasi rigid semimorphisms, etc.

Rectangulotopes

Following the notions of Ω-set and Ω-algebra where Ω is a complete lattice, we introduce P-algebras, replacing the lattice Ω by a poset P. A P-algebra is a classical algebraic structure in which the usual equality is replaced by a P-valued equivalence relation, i.e., with the symmetric and transitive map from the underlying set into a poset P. In addition, this generalized equality is (as a map) compatible with the fundamental operations of the algebra. The diagonal restriction of this map is a P-valued support of a P-algebra. The particular subsets of this support, its cuts, are classical subalgebras, while the cuts of the P-valued equality are congruences on the corresponding cut subalgebras. We prove that the collection of the corresponding quotients of these cuts is a centralized system in the lattice of weak congruences of the basic algebra. We also describe the canonical representation of P-algebras, independent of the poset P.

The height of a poset P is the supremum of the cardinalities of chains in P. The exact formula for the height of the subgroup lattice of the symmetric group S n is known, as is an accurate asymptotic formula for the height of the subsemigroup lattice of the full transformation monoid T n . Motivated by the related question of determining the heights of the lattices of left and right congruences of T n , and deploying the framework of unary algebras and semigroup actions, we develop a general method for computing the heights of lattices of both one-and two-sided congruences for semigroups. We apply this theory to obtain exact height formulae for several monoids of transformations, matrices and partitions, including the full transformation monoid T n , the partial transformation monoid PT n , the symmetric inverse monoid I n , the monoid of order-preserving transformations O n , the full matrix monoid M(n, q), the partition monoid P n , the Brauer monoid B n and the Temperley-Lieb monoid T L n .

Subresiduated Nelson algebras

We show that there are finite distributive lattices that are not the congruence lattice of any finite semidistributive lattice. For \(0 \leq k \leq 2\), the distributive lattice \((\mathbf{B}_k)_{++} = \mathbf{2} + \mathbf{B}_k\), where \(\mathbf{B}_k\) denotes the boolean lattice with \(k\) atoms, is not the congruence lattice of any finite semidistributive lattice. Neither can these lattices be a filter in the congruence lattice of a finite semidistributive lattice. However, each \((\mathbf{B}_k)_{++}\) with \(k \geq 3\) is the congruence lattice of a finite semidistributive lattice, say \(\mathbf{L}_k\). These lattices \(\mathbf{L}_k\) cannot be bounded (in the sense of McKenzie), as no \((\mathbf{B}_k)_{++}\) \((k \geq 0)\) is the congruence lattice of a finite bounded lattice. A companion paper shows that every \((\mathbf{B}_k)_{++}\) \((k \geq 0)\) can be represented as the congruence lattice of an infinite semidistributive lattice. We also find sufficient conditions for a finite distributive lattice to be representable as the congruence lattice of a finite bounded (and hence semidistributive) lattice.

In this paper we study properties of congruence lattices of algebras with one operator and the main symmetric operation. A ternary operation 𝑑(𝑥, 𝑦, 𝑧) satisfying identities 𝑑(𝑥, 𝑦, 𝑦) = 𝑑(𝑦, 𝑦, 𝑥) = 𝑑(𝑦,𝑥,𝑦)= 𝑥 is called a minority operation. The symmetric operation is aminority operation defined by specific way. An algebra 𝐴 is called a chain algebra if 𝐴 has a linearly ordered congruence lattice. An algebra 𝐴 is called subdirectly irreducible if 𝐴 has the smallest nonzero congruence. An algebra with operators is an universal algebra whose signature consists of two nonempty non-intersectional parts: the main part which can contain arbitrary operations, and the additional part consisting of operators. The operators are unary operations that act as endomorphisms with respect to the main operations, i.e., one are permutable with the main operations. An unar is an algebra with one unary operation. If 𝑓 is the unary operation from the signature Ω then the unar ⟨𝐴, 𝑓⟩ is called an unary reduct of algebra ⟨𝐴,Ω⟩.A description of algebras with one operator and the main symmetric operation that have a linear ordered congruence lattice is obtained. It shown that algebra of given class is a chain algebra if and only if one is subdirectly irreducible. For algebras of given class we obtained necessary and sufficient conditions for the coincidence of their congruence lattices and congruence lattices of unary reducts these algebras.

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

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.

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.

The lattice of quasi-orders of the universal algebra 𝐴 is the lattice of those quasi-orders on the set 𝐴 that are compatible with the operations of the algebra, the lattice of the topologies of the algebra is the lattice of those topologies with respect to which the operations of the algebra are continuous. The lattice of quasi-orders and the lattice of topologies of the algebra 𝐴, along withthe lattice of subalgebras and the lattice of congruences, are important characteristics of this algebra. It is known that a lattice of quasi-orders is isomorphically embedded in a lattice that is anti-isomorphic to a lattice of topologies, and in the case of a finite algebra, this embedding is an anti-isomorphism. A chain 𝑋𝑛 of 𝑛 elements is considered as a lattice with operations𝑥 ∧ 𝑦 = min(𝑥, 𝑦) and 𝑥 ∨ 𝑦 = max(𝑥, 𝑦). It is proved that the lattice of quasi-orders and the lattice of topologies of the chain 𝑋𝑛 are isomorphic to the Boolean lattice of 2^(2𝑛−2) elements. A simple correspondence is found between the quasi-orders of the chain 𝑋𝑛 and words of length 𝑛 − 1 in a 4-letter alphabet. Atoms of the lattice of topologies are found. We deduce from the results on quasi-orders a well-known statement that the congruence lattice of an 𝑛-element chain is Boolean lattioce of 2^(𝑛−1) elements. The results will no longer be true if the chain is considered only with respect to one of the operations ∧,∨.

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.

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.

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.

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.