Congruence Lattice Research Articles

On prevarieties of logic

It is proved that every prevariety of algebras is categorically equivalent to a ‘prevariety of logic’, i.e., to the equivalent algebraic semantics of some sentential deductive system. This allows us to show that no nontrivial equation in the language $$\wedge ,\vee ,\circ $$ holds in the congruence lattices of all members of every variety of logic, and that being a (pre)variety of logic is not a categorical property.

On the variety of Gödel MV-algebras

We introduce a new algebraic structure $$\begin{aligned} (A, \otimes , \oplus , *, \vee , \wedge , \rightharpoonup , 0, 1) \end{aligned}$$ called Godel–MV-algebra (GMV-algebra) such that It is shown that the lattice of congruences of a GMV -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 $$(A, \otimes , \oplus , *, 0, 1)$$ . Any GMV-algebra is bi-Heyting algebra. Any chain GMV-algebra is simple, and any GMV-algebra is semi-simple. Finitely generated GMV-algebras are described, and finitely generated finitely presented GMV-algebras are characterized. The algebraic counterpart of axiomatically presented GMV-logic is GMV-algebras .

Minimal representations of a finite distributive lattice by principal congruences of a lattice

Let the finite distributive lattice $D$ be isomorphic to the congruence lattice of a finite lattice $L$. Let $Q$ denote those elements of $D$ that correspond to principal congruences under this isomorphism. Then $Q$ contains $0,1 \in D$ and all the join-irreducible elements of $D$. If $Q$ contains exactly these elements, we say that $L$ is a minimal representations of $D$ by principal congruences of the lattice $L$. We characterize finite distributive lattices $D$ with a minimal representation by principal congruences with the property that $D$ has at most two dual atoms.

Partial frames and filter spaces

In general topology, filters have proved a popular tool for the construction of completions. In pointfree topology, downsets are frequently employed for the same purpose. Indeed, any downset frame can be viewed as the topology of the space of filters on that frame. In this paper we consider such ideas in the setting of partial frames; these are meet-semilattices where, in contrast with frames, not all subsets need have joins; a so-called selection function specifies those subsets that must have joins. We have made use of the lattice of certain kinds of downsets in several situations; to understand the spatial aspect of these downsets, we are led naturally to consider the notion of partial spaces as a generalization of topological spaces. We establish an adjunction between partial frames and partial spaces which restricts to an equivalence between spatial partial frames and sober partial spaces, and then obtain the desired result that the appropriate collection of downsets is isomorphic to the collection of opens of an appropriate filter space. Both the filter space and the downset partial frame construction are functorial; and the two functors are linked by two natural isomorphisms that involve the open and spectrum functors for partial spaces and partial frames.Next we consider questions of spatiality for the congruence frame of a partial frame. Quotients for frames can be provided using nuclei or congruences but in the setting of partial frames nuclei do not suffice, whereas congruences do. Whether the congruence lattice of a partial frame is itself a frame or not depends on the axioms assumed for the selection function under consideration: if at least all finite subsets are selected, the congruence lattice is indeed a frame. For such, we characterize the spatiality of the frame of all congruences on a partial frame in terms of its quotients.

On representation of finite lattices

In an earlier article, the authors found sufficient conditions for complexity of the lattice of subquasivarieties of a quasivariety. In the present article, we prove that these conditions allow us to represent finite lattices as relative congruence lattices and relative variety lattices in a uniform way. Some applications are presented.

Quotientopes

For any lattice congruence of the weak order on $\mathfrak{S}_n$, N. Reading proved that glueing together the cones of the braid fan that belong to the same congruence class defines a complete fan. We prove that this fan is the normal fan of a polytope.

The largest higher commutator sequence

Given the congruence lattice L of a finite algebra A that generates a congruence permutable variety, we look for those sequences of operations on L that have the properties of higher commutator operations of expansions of A. If we introduce the order of such sequences in the natural way the question is whether exists or not the largest one. The answer is positive. We provide a description of the largest element and as a consequence we obtain that the sequences form a complete lattice.

On the set of principal congruences in a distributive congruence lattice of an algebra

Let $Q$ be a subset of a finite distributive lattice $D$. An algebra $A$ represents the inclusion $Q\subseteq D$ by principal congruences if the congruence lattice of $A$ is isomorphic to $D$ and the ordered set of principal congruences of $A$ corresponds to $Q$ under this isomorphism. If there is such an algebra for every subset $Q$ containing $0$, $1$, and all join-irreducible elements of $D$, then $D$ is said to be (A1)-representable. We prove that every (A1)-representable finite distributive lattice is planar and it has at most one join-reducible coatom. Conversely, we prove that every finite planar distributive lattice with at most one join-reducible coatom is chain-representable in the sense of a recent paper of G. Gratzer. Combining the results of this paper with another paper by the present author, it follows that every (A1)-representable finite distributive lattice is fully representable even by principal congruences of finite lattices. Finally, we prove that every chain-representable inclusion $Q\subseteq D$ can be represented by the principal congruences of a finite (and quite small) algebra.

Meet-irreducible congruence lattices

The system of all congruences of an algebra (A, F) forms a lattice, denoted $${{\mathrm{Con}}}(A, F)$$ . Further, the system of all congruence lattices of all algebras with the base set A forms a lattice $$\mathcal {E}_A$$ . We deal with meet-irreducibility in $$\mathcal {E}_A$$ for a given finite set A. All meet-irreducible elements of $$\mathcal {E}_A$$ are congruence lattices of monounary algebras. Some types of meet-irreducible congruence lattices were described in Jakubikova-Studenovska et al. (2017). In this paper, we prove necessary and sufficient conditions under which $${{\mathrm{Con}}}(A, f)$$ is meet-irreducible in the case when (A, f) is an algebra with short tails (i.e., f(x) is cyclic for each $$x \in A$$ ) and in the case when (A, f) is an algebra with small cycles (every cycle contains at most two elements).

Congruences on near-Heyting algebras

A near-Heyting algebra is a join-semilattice with a top element such that every principal upset is a Heyting algebra. We establish a one-to-one correspondence between the lattices of filters and congruences of a near-Heyting algebra. To attain this aim, we first show an embedding from the lattice of filters to the lattice of congruences of a distributive nearlattice. Then, we describe the subdirectly irreducible and simple near-Heyting algebras. Finally, we fully characterize the principal congruences of distributive nearlattices and near-Heyting algebras. We conclude that the varieties of distributive nearlattices and near-Heyting algebras have equationally definable principal congruences.

Hopf algebras on decorated noncrossing arc diagrams

Noncrossing arc diagrams are combinatorial models for the equivalence classes of the lattice congruences of the weak order on permutations. In this paper, we provide a general method to endow these objects with Hopf algebra structures. Specific instances of this method produce relevant Hopf algebras that appeared earlier in the literature.

OVERGROUPS OF WEAK SECOND MAXIMAL SUBGROUPS

A subgroup $H$ is called a weak second maximal subgroup of $G$ if $H$ is a maximal subgroup of a maximal subgroup of $G$. Let $m(G,H)$ denote the number of maximal subgroups of $G$ containing $H$. We prove that $m(G,H)-1$ divides the index of some maximal subgroup of $G$ when $H$ is a weak second maximal subgroup of $G$. This partially answers a question of Flavell [‘Overgroups of second maximal subgroups’, Arch. Math.64(4) (1995), 277–282] and extends a result of Pálfy and Pudlák [‘Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups’, Algebra Universalis11(1) (1980), 22–27].

Cometic functors and representing order-preserving maps by principal lattice congruences

Let $$\mathbf {Lat}^{\mathrm{sd}}_{5}$$ and $$\mathbf {Pos}_{01}^{{\scriptscriptstyle \mathrm{{+}}}}$$ denote the category of selfdual bounded lattices of length 5 with $$\{0,1\}$$ -preserving lattice homomorphisms and that of bounded ordered sets with $$\{0,1\}$$ -preserving isotone maps, respectively. For an object L in $$\mathbf {Lat}^{\mathrm{sd}}_{5}$$ , the ordered set of principal congruences of the lattice L is denoted by $$\mathrm{Princ}(L)$$ . By means of congruence generation, $$\mathrm{Princ}:\mathbf {Lat}^{\mathrm{sd}}_{5}\rightarrow \mathbf {Pos}_{01}^{{\scriptscriptstyle \mathrm{{+}}}}$$ is a functor. We prove that if $$\mathbf {A}$$ is a small subcategory of $$\mathbf {Pos}_{01}^{{\scriptscriptstyle \mathrm{{+}}}}$$ such that every morphism of $$\mathbf {A}$$ is a monomorphism, understood in $$\mathbf {A}$$ , then $$\mathbf {A}$$ is the $$\mathrm{Princ}$$ -image of an appropriate subcategory of $$\mathbf {Lat}^{\mathrm{sd}}_{5}$$ . This result extends G. Gratzer’s earlier theorems where $$\mathbf {A}$$ consisted of one or two objects and at most one non-identity morphism, and the author’s earlier result where all morphisms of $$\mathbf {A}$$ were 0-separating and no hom-set had more the two morphisms. Furthermore, as an auxiliary tool, we derive some families of maps, also known as functions, from injective maps and surjective maps; this can be useful in various fields of mathematics, not only in lattice theory. Namely, for every small concrete category $$\mathbf {A}$$ , we define a functor $${F_{{\scriptscriptstyle \mathrm{{com}}}}}$$ , called cometic functor, from $$\mathbf {A}$$ to the category $$\mathbf Set $$ of sets and a natural transformation $${{\varvec{\pi }}^{{\scriptscriptstyle \mathrm{{com}}}}}$$ , called cometic projection, from $${F_{{\scriptscriptstyle \mathrm{{com}}}}}$$ to the forgetful functor of $$\mathbf {A}$$ into $$\mathbf Set $$ such that the $${F_{{\scriptscriptstyle \mathrm{{com}}}}}$$ -image of every monomorphism of $$\mathbf {A}$$ is an injective map and the components of $${{\varvec{\pi }}^{{\scriptscriptstyle \mathrm{{com}}}}}$$ are surjective maps.

Finite Semilattices with Many Congruences

For an integer n ≥ 2, let NCSL(n) denote the set of sizes of congruence lattices of n-element semilattices. We find the four largest numbers belonging to NCSL(n), provided that n is large enough to ensure that |NCSL(n)|≥ 4. Furthermore, we describe the n-element semilattices witnessing these numbers.

Semirings of Continuous (0,∞]-Valued Functions

The semiring C∞(X) of all continuous functions on an arbitrary topological space X with values in the topological semiring (0,∞] is studied. General properties of semirings C∞(X) are considered. Properties of lattices of ideals and congruences of semirings C∞(X) over the P-spaces X, the F-spaces X, and the finite discrete spaces are proved.

Representing an isotone map between two bounded ordered sets by principal lattice congruences

For bounded lattices L1 and L2, let $${f\colon L_1 \to L_2}$$ be a lattice homomorphism. Then the map $${{\rm Princ}(f)\colon \rm {Princ}(\it L_1) \to {\rm Princ}(\it L_2)}$$ , defined by $${{\rm con}(x,y) \mapsto {\rm con}(f(x),f(y))}$$ , is a 0-preserving isotone map from the bounded ordered set Princ(L1) of principal congruences of L1 to that of L2. We prove that every 0-preserving isotone map between two bounded ordered sets can be represented in this way. Our result generalizes a 2016 result of G. Gratzer from $${\{0,1}\}$$ -preserving isotone maps to 0-preserving isotone maps.

Congruence lattices of finite diagram monoids

We give a complete description of the congruence lattices of the following finite diagram monoids: the partition monoid, the planar partition monoid, the Brauer monoid, the Jones monoid (also known as the Temperley–Lieb monoid), the Motzkin monoid, and the partial Brauer monoid. All the congruences under discussion arise as special instances of a new construction, involving an ideal I, a retraction I→M onto the minimal ideal, a congruence on M, and a normal subgroup of a maximal subgroup outside I.

On Monadic Operators on Modal Pseudocomplemented De Morgan Algebras and Tetravalent Modal Algebras

In our paper, monadic modal pseudocomplemented De Morgan algebras (or mmpM) are considered following Halmos’ studies on monadic Boolean algebras. Hence, their topological representation theory (Halmos–Priestley’s duality) is used successfully. Lattice congruences of an mmpM is characterized and the variety of mmpMs is proven semisimple via topological representation. Furthermore and among other things, the poset of principal congruences is investigated and proven to be a Boolean algebra; therefore, every principal congruence is a Boolean congruence. All these conclusions contrast sharply with known results for monadic De Morgan algebras. Finally, we show that the above results for mmpM are verified for monadic tetravalent modal algebras.

Lattice structure of Weyl groups via representation theory of preprojective algebras

This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$, using representation theory of the corresponding preprojective algebra $\unicode[STIX]{x1D6F1}$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $\unicode[STIX]{x1D70F}$-rigid (respectively, $\unicode[STIX]{x1D70F}^{-}$-rigid) modules and layers of $\unicode[STIX]{x1D6F1}$. The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\unicode[STIX]{x1D6F1}$. We show that layers of $\unicode[STIX]{x1D6F1}$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton extension order. We give a combinatorial description of indecomposable $\unicode[STIX]{x1D70F}^{-}$-rigid modules for type $A$ and $D$.

Quasiorder lattices of varieties

The set $${{\mathrm{Quo}}}(\mathbf {A})$$ of compatible quasiorders (reflexive and transitive relations) of an algebra $$\mathbf {A}$$ forms a lattice under inclusion, and the lattice $${{\mathrm{Con}}}(\mathbf {A})$$ of congruences of $$\mathbf {A}$$ is a sublattice of $${{\mathrm{Quo}}}(\mathbf {A})$$ . We study how the shape of congruence lattices of algebras in a variety determine the shape of quasiorder lattices in the variety. In particular, we prove that a locally finite variety is congruence distributive [modular] if and only if it is quasiorder distributive [modular]. We show that the same property does not hold for meet semi-distributivity. From tame congruence theory we know that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of congruence lattices isomorphic to the lattice $$\mathbf {M}_3$$ . We prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for quasiorder lattices of infinite algebras even in the variety of semilattices.