A subclass of Ockham algebras

Here we introduce a subclass of the class of Ockham algebras ( L ; f ) for which L satisfies the property that for every x ∈ L , there exists n ≥ 0 such that fn ( x ) and fn+1 ( x ) are complementary. We characterize the structure of the lattice of congruences on such an algebra ( L ; f ). We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.     

Homomorphisms and principal congruences of bounded lattices. II. Sketching the proof for sublattices

A recent result of G. Czédli relates the ordered set of principal congruences of a bounded lattice L with the ordered set of principal congruences of a bounded sublattice K of L. In this note, I sketch a new proof.     

A note on representation of lattices by weak congruences

A weak congruence is a symmetric, transitive, and compatible relation. An element u of an algebraic lattice L is ??-suitable if there is an isomorphism ?? from L to the lattice of weak congruences of an algebra such that ??(u) is the diagonal relation. Some conditions implying the ??-suitability of u are presented.     

Representing some families of monotone maps by principal lattice congruences

For a lattice L with 0 and 1, let Princ(L) denote the set of principal congruences of L. Ordered by set inclusion, it is a bounded ordered set. In 2013, G. Grätzer proved that every bounded ordered set is representable as Princ(L); in fact, he constructed L as a lattice of length 5. For {0, 1}-sublattices \({A \subseteq B}\) of L, congruence generation defines a natural map Princ(A) \({\longrightarrow}\) Princ(B). In this way, every family of {0, 1}-sublattices of L yields a small category of bounded ordered sets as objects and certain 0-separating {0, 1}-preserving monotone maps as morphisms such that every hom-set consists of at most one morphism. We prove the converse: every small category of bounded ordered sets with these properties is representable by principal congruences of selfdual lattices of length 5 in the above sense. As a corollary, we can construct a selfdual lattice L in G. Grätzer's above-mentioned result.     

A proof of Frankl’s union-closed sets conjecture for dismantlable lattices

For a lattice L, let Princ(L) denote the ordered set of principal congruences of L. In a pioneering paper, G. Grätzer characterized the ordered set Princ(L) of a finite lattice L; here we do the same for a countable lattice. He also showed that every bounded ordered set H is isomorphic to Princ(L) of a bounded lattice L. We prove a related statement: if an ordered set H with a least element is the union of a chain of principal ideals (equivalently, if 0 \({\in}\) H and H has a cofinal chain), then H is isomorphic to Princ(L) of some lattice L.     

∗-congruences on abundant semigroups with SQ-adequate transversals

We give ?-congruences on an abundant semigroup with an SQ-adequate transversal S ^° by the ?-congruence triple abstractly which consists of congruences on the structure component parts L, T and R. We prove that the set of all ?-congruences on this kind of semigroups is a complete lattice.     

Homomorphisms and principal congruences of bounded lattices. III. The Independence Theorem

A new result of G. Czédli states that for an ordered set P with at least two elements and a group G, there exists a bounded lattice L such that the ordered set of principal congruences of L is isomorphic to P and the automorphism group of L is isomorphic to G.I provide an alternative proof utilizing a result of mine with J. Sichler from the late 1990s.     

A problem on bands

In this paper, we consider an open problem proposed by Petrich and Reilly: What are necessary and sufficient conditions on a completely regular semigroup S in order that the trace relation T on the lattice of congruences on S is equal to the identity relation? By constructing some special congruences on S, we prove that T=ε if and only if S is a band.     

Congruences on ockham algebras with pseudocomplementation

The variety pOconsists of those algebras (L;?,?,f,^*,0,1) where (L;?,?,f,0,1) is an Ockham algebra, (L;?,?,f,^*,0,1) is a p-algebra, and the unary operations fand ^*. commute. For an algebra in pK _ωwe show that the compact congruences form a dual Stone lattice and use this to determine necessary and sufficient conditions for a principal congruence to be complemented. We also describe the lattice of subvarieties of pK _1,1identifying therein the biggest subvariety in which every principal congruence is complemented, and the biggest subvariety in which the intersection of two principal congruences is principal.     

Finite distributive lattices are congruence lattices of almost-geometric lattices

A semimodular lattice L of finite length will be called an almost-geometric lattice if the order J(L) of its nonzero join-irreducible elements is a cardinal sum of at most two-element chains. We prove that each finite distributive lattice is isomorphic to the lattice of congruences of a finite almost-geometric lattice.     

Characterizing fully principal congruence representable distributive lattices

Motivated by a recent paper of G. Grätzer, a finite distributive lattice D is called fully principal congruence representable if for every subset Q of D containing 0, 1, and the set J(D) of nonzero join-irreducible elements of D, there exists a finite lattice L and an isomorphism from the congruence lattice of L onto D such that Q corresponds to the set of principal congruences of L under this isomorphism. A separate paper of the present author contains a necessary condition of full principal congruence representability: D should be planar with at most one join-reducible coatom. Here we prove that this condition is sufficient. Furthermore, even the automorphism group of L can arbitrarily be stipulated in this case. Also, we generalize a recent result of G. Grätzer on principal congruence representable subsets of a distributive lattice whose top element is join-irreducible by proving that the automorphism group of the lattice we construct can be arbitrary.     

Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices

Let be a {0, 1}-homomorphism of a finite distributive lattice D into the congruence lattice Con L of a rectangular (whence finite, planar, and semimodular) lattice L. We prove that L is a filter of an appropriate rectangular lattice K such that ConK is isomorphic with D and is represented by the restriction map from Con K to Con L. The particular case where is an embedding was proved by E.T. Schmidt. Our result implies that each {0, 1}-lattice homomorphism between two finite distributive lattices can be represented by the restriction of congruences of an appropriate rectangular lattice to a rectangular filter.     

On the tolerance lattice of tolerance factors

Although the notion of a tolerance is a natural generalization of the notion of a congruence, many properties of factor lattices modulo congruences are not, in general, valid for factor lattices modulo tolerances. In this paper, for a lattice L of a finite length, we define a new partial order ? on $\operatorname{Tol}\, (L)$ such that for every ${S\in \operatorname{Tol}\, (L)}$ with T?S, a tolerance S/T is induced on the factor lattice L/T. This partial order is a particular restriction of ? and thus we can prove for tolerances some analogous results to the homomorphism theorem and the second isomorphism theorem for congruences. The poset $(\operatorname{Tol}\, (L), \sqsubseteq)$ is not always a lattice, but it can be converted into a specific commutative join-directoid. Then, for every ${T\in \operatorname{Tol}\, (L)}$ , $(\operatorname{Tol}\, (L/T),\sqsubseteq)$ constitutes a subdirectoid of the directoid based on the poset $(\operatorname{Tol}\, (L),\sqsubseteq)$ and this specific directoid structure is preserved by the direct product of lattices.     

On the number of join irreducibles and acyclicity in finite modular lattices

Bases of lines provide useful presentations of finite height modular lattices, acyclic ones being related to amenable properties in equational and representation theory. It is shown that some (equivalently: any) base of L is acyclic if and only if L has exactly 2d(L)?s(L) join irreducibles; moreover, that this is the minimal possible number for any L. Here d(L) denotes the height and s(L) the number of maximal congruences of L.     

On Hopf Algebras with Nonzero Integral

Let S be the model of a semigroup with an associate subgroup whose identity is a medial idempotent constructed by Blyth and Martins considered as a unary semigroup. For another such semigroup T, we construct all unary homomorphisms of S into T in terms of their parameters. On S we construct all unary congruences again directly from its parameters. This construction leads to a characterization of congruences in terms of kernels and traces. We describe the K, T, L, U and V relations on the lattice of all unary congruences on S, again in terms of parameters of S.     

Über Homologiegruppen von Faserbündeln

Let L be a distributive lattice with 0 and C (L) be its lattice of congruences. The skeleton, SC (L), of C (L) consists of all those congruences which are the pseudocomplements of members of C (L), and is a complete BOOLEan lattice. An ideal is the kernel of a skeletal congruence if and only if it is an intersection of relative annihilator ideals, i.e. ideals of the form <r, s>j={x∈L: xΔr≤s} for suitable r, s∈L. The set KSC (L) of all such kernels forms an upper continuous distributive lattice and the map a ? (a={x∈L: x≤a} is a lower regular joindense embedding of L into KSC (L). The relationship between SC (L) and KSC (L) leads to numerous characterizations of disjunctive and generalized BOOLEan lattices. In particular, a distributive lattice L is disjunctive (generalized Boolean) if and only if the map Θ ? ker Θ is a lattice-isomorphism of SC (L) onto KSC (L), whose inverse is the map J ? Θ (J)** (the map J ? Θ(J)). In addition, a study of KSC (L) leads to new simple proofs of results on the completions of special classes of lattices.     

Congruence kernels in Ockham algebras

We consider, in the context of an Ockham algebra \({{\mathcal{L} = (L; f)}}\), the ideals I of L that are kernels of congruences on \({\mathcal{L}}\). We describe the smallest and the largest congruences having a given kernel ideal, and show that every congruence kernel \({I \neq L}\) is the intersection of the prime ideals P such that \({I \subseteq P}\), \({P \cap f(I) = \emptyset}\), and \({f^{2}(I) \subseteq P}\). The congruence kernels form a complete lattice which in general is not modular. For every non-empty subset X of L, we also describe the smallest congruence kernel to contain X, in terms of which we obtain necessary and sufficient conditions for modularity and distributivity. The case where L is of finite length is highlighted.     

Congruences via abelian groups

Given a group G acting on a set S, Möbius inversion over the lattice of subgroups can be used to obtain congruences relating the number of elements of S stabilized by each subgroup. By taking S to be a set of subsets, partitions, or permutations congruences for binomial and multinomial coefficients, Stirling numbers of both kinds, and various other combinatorial sequences are derived. Congruences for different moduli are obtained by varying the order of G.     

Lattice uniformities on pseudo-D-lattices

Let L be a pseudo-D-lattice. We prove that the lattice uniformities on L which make uniformly continuous the operations of L are uniquely determined by their system of neighbourhoods of 0 and form a distributive lattice. Moreover we prove that every such uniformity is generated by a family of weakly subadditive [0,+∞]-valued functions on L.     

On the lattice of stratified principal L-topologies

We investigate the lattice structure of the set of all stratified principal L-topologies on a given set X. It proves that the lattice of stratified principal L-topologies S p(X) has atoms and dual atoms if and only if L has atoms and dual atoms respectively. Moreover, it is complete and semi-complemented. We also discuss some other properties of the lattice.