Separation in distributive congruence lattices

We define separable sets in algebraic lattices. For a finitely generated congruence distributive variety V \mathcal{V} , we show a close connection between non-separable sets in congruence lattices of algebras in V \mathcal{V} and the structure of subdirectly irreducible algebras in V \mathcal{V} . We apply the general results to some lattice varieties.     

Complete congruence lattices of join-infinite distributive lattices

Received July 26, 1993; accepted in final form July 16, 1996.     

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.     

On complete congruence lattices of join-infinite distributive lattices

In [5] G. Gr?tzer and E. T. Schmidt raised the problem of characterizing the complete congruence lattices of complete lattices satisfying the Join-Infinite Distributive Identity (JID) and the Meet-Infinite Distributive Identity (MID) and proved the theorem: Any complete lattice with more than two elements and with a meet-irreducible zero cannot be represented as the lattice of complete congruence relations of a complete lattice satisfying the (JID) and (MID). In this note we generalize this result by showing that the complete congruence lattice of every complete lattice satisfying (JID) and (MID) is a zero-dimensional complete lattice satisfying (JID). Some consequences are discussed. Received March 6, 2000; accepted in final form September 12, 2000.     

Relative separation in distributive congruence lattices

In [5] we defined separable sets in algebraic lattices and showed a close connection between the types of non-separable sets in congruence lattices of algebras in a finitely generated congruence distributive variety and the structure of subdirectly irreducible algebras in Now we generalize these results using the concept of relatively separable sets (with respect to subsets) and apply them to some lattice varieties.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived November 29, 2002; accepted in final form August 19, 2004.     

Iterative separation in distributive congruence lattices

In [PLOŠČICA, M.: Separation in distributive congruence lattices, Algebra Universalis 49 (2003), 1–12] we defined separable sets in algebraic lattices and showed a close connection between the types of non-separable sets in congruence lattices of algebras in a finitely generated congruence distributive variety and the structure of subdirectly irreducible algebras in . Now we generalize these results using the concept of separable mappings (defined on some trees) and apply them to some lattice varieties. Supported by VEGA Grants 2/4134/24, 2/7141/27, and INTAS Grant 03-51-4110.     

Definable principal congruences in congruence distributive varieties

We present an algorithm that, given a finite algebraA generating a congruence distributive (CD) variety, determines whether this variety has first order definable principal congruences (DPC). In fact, DPC turns out to be equivalent to the extendability of the principal congruences of certain subalgebras of the algebras in HS(A ³). To verify this algorithm, we investigate combinatorial properties of finite subdirect powers ofA. Our theorem has a relatively simple formulation for arithmetical algebras. As an application, we obtain McKenzie's result that there are no nondistributive lattice varieties with DPC.Presented by A. Pixley.Finally I wish to thank E. Fried, R. W. Quackenbush and P. Pröhle for many helpful conversations (some ideas of the paper came up by considering weakly associative lattices), and to A. F. Pixley for raising the problem mentioned in the first section, which was the starting point of this investigation.     

Completely representable lattices

It is known that a lattice is representable as a ring of sets iff the lattice is distributive. CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets. jCRL is the class of DLs which have representations preserving arbitrary joins, mCRL is the class of DLs which have representations preserving arbitrary meets, and biCRL is defined to be \({{\bf jCRL} \cap {\bf mCRL}}\) . We prove

${\bf CRL} \subset {\bf biCRL} = {\bf mCRL} \cap {\bf jCRL} \subset {\bf mCRL} \neq {\bf jCRL} \subset {\bf DL}$

where the marked inclusions are proper.

Let L be a DL. Then \({L \in {\bf mCRL}}\) iff L has a distinguishing set of complete, prime filters. Similarly, \({L \in {\bf jCRL}}\) iff L has a distinguishing set of completely prime filters, and \({L \in {\bf CRL}}\) iff L has a distinguishing set of complete, completely prime filters.Each of the classes above is shown to be pseudo-elementary, hence closed under ultraproducts. The class CRL is not closed under elementary equivalence, hence it is not elementary.     

The FEP for some varieties of fully distributive knotted residuated lattices

We prove the finite embeddability property for a wide range of varieties of fully distributive residuated lattices and FL-algebras. Part of the axiomatization is assumed to be a knotted inequality and some appropriate generalization of commutativity. The construction is based on distributive residuated frames and extends to subvarieties axiomatized by any division-free equation.     

Endoprimal distributive lattices

Presented by A. Szendrei.     

Idempotent matrix lattices over distributive lattices

In this paper, the partially ordered set of idempotent matrices over distributive lattices with the partial order induced by a set of lattice matrices is studied. It is proved that this set is a lattice; the formulas for meet and join calculation are obtained. In the lattice of idempotent matrices over a finite distributive lattice, all atoms and coatoms are described. We prove that the lattice of quasi-orders over an n-element set Qord(n) is not graduated for n ≥ 3 and calculate the greatest and least lengths of maximal chains in this lattice. We also prove that the interval ([I, J]_≤, ≤) of idempotent (n × n)-matrices over {ie879-01}-lattices is isomorphic to the lattice of quasi-orders Qord(n). Using this isomorphism, we calculate the lattice height of idempotent {ie879-02}-matrices. We obtain a structural criterion of idempotent matrices over distributive lattices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 121–144, 2007.     

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 Jakubíková-Studenovská 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).     

Computing congruence lattices of finite lattices

An inequality between the number of coverings in the ordered set of join irreducible congruences on a lattice and the size of is given. Using this inequality it is shown that this ordered set can be computed in time , where .