Two More Topics on Congruence Lattices of Lattices

2014 ◽  
pp. 393-435 ◽  
Author(s):  
George Grätzer
Keyword(s):  
Congruence Lattices

scholarly journals Congruence pairs of principal MS-algebras and perfect extensions

2020 ◽  
Vol 70 (6) ◽  
pp. 1275-1288
Author(s):  
Abd El-Mohsen Badawy ◽  
Miroslav Haviar ◽  
Miroslav Ploščica
Keyword(s):  
Boolean Algebra ◽  
Congruence Lattices ◽  
Descriptive Solution ◽  
The One ◽  
Special Case ◽  
Congruence Pair

AbstractThe notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for K2-algebras [6], is introduced. It is proved that the congruences of the principal MS-algebras L correspond to the MS-congruence pairs on simpler substructures L°° and D(L) of L that were associated to L in [4].An analogy of a well-known Grätzer’s problem [11: Problem 57] formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in [2], it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant.As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra L is a perfect extension of its greatest Stone subalgebra LS. It is shown that this is exactly when de Morgan subalgebra L°° of L is a perfect extension of the Boolean algebra B(L). Two examples illustrating when this special case happens and when it does not are presented.

scholarly journals CONGRUENCE FD-MAXIMAL VARIETIES OF ALGEBRAS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250053 ◽  
Author(s):  
PIERRE GILLIBERT ◽  
MIROSLAV PLOŠČICA
Keyword(s):  
Necessary Condition ◽  
Finitely Generated ◽  
Distributive Variety ◽  
Finite Lattices ◽  
Congruence Lattices ◽  
Congruence Distributive Variety ◽  
Congruence Distributive ◽  
Special Congruence

We study the class of finite lattices that are isomorphic to the congruence lattices of algebras from a given finitely generated congruence-distributive variety. If this class is as large as allowed by an obvious necessary condition, the variety is called congruence FD-maximal. The main results of this paper characterize some special congruence FD-maximal varieties.

scholarly journals Congruence Representations of Join-homomorphisms of Finite Distributive Lattices: Size and Breadth

2000 ◽  
Vol 68 (1) ◽  
pp. 85-103 ◽  
Author(s):  
G. Grätzer ◽  
H. Lakser ◽  
E. T. Schmidt
Keyword(s):  
Distributive Lattices ◽  
Irreducible Elements ◽  
Finite Lattices ◽  
Congruence Lattices ◽  
Finite Distributive Lattices

AbstractLet K and L be lattices, and let ϕ be a homomorphism of K into L.Then ϕ induces a natural 0-preserving join-homomorphism of Con K into Con L.Extending a result of Huhn, the authors proved that if D and E are finite distributive lattices and ψ is a 0-preserving join-homomorphism from D into E, then D and E can be represented as the congruence lattices of the finite lattices K and L, respectively, such that ψ is the natural 0-preserving join-homomorphism induced by a suitable homomorphism ϕ: K → L. Let m and n denote the number of join-irreducible elements of D and E, respectively, and let k = max (m, n). The lattice L constructed was of size O(22(n+m)) and of breadth n+m.We prove that K and L can be constructed as ‘small’ lattices of size O(k5) and of breadth three.

scholarly journals Congruence Lattices of Finite Semimodular Lattices

1998 ◽  
Vol 41 (3) ◽  
pp. 290-297 ◽  
Author(s):  
G. Grätzer ◽  
H. Lakser ◽  
E. T. Schmidt
Keyword(s):  
Distributive Lattice ◽  
Congruence Lattice ◽  
Finite Distributive Lattice ◽  
Congruence Lattices ◽  
Semimodular Lattices

AbstractWe prove that every finite distributive lattice can be represented as the congruence lattice of a finite (planar) semimodular lattice.

Morita invariants of semirings

2015 ◽  
Vol 15 (02) ◽  
pp. 1650023 ◽  
Author(s):  
Sujit Kumar Sardar ◽  
Sugato Gupta
Keyword(s):  
Morita Equivalence ◽  
Ideal Lattices ◽  
Congruence Lattices

In this paper we revisit that ideal lattices and congruence lattices are preserved by Morita equivalence of semirings which is originally obtained implicitly by Katsov and his co-authors. This is then used to obtain some Morita invariants for semirings.

scholarly journals Congruence lattices of semilattices

1973 ◽  
Vol 49 (1) ◽  
pp. 51-58 ◽  
Author(s):  
Ralph Freese ◽  
James Nation
Keyword(s):  
Congruence Lattices

scholarly journals A note on atomistic weak congruence lattices

2008 ◽  
Vol 308 (10) ◽  
pp. 2054-2057 ◽  
Author(s):  
Branimir Šešelja ◽  
Andreja Tepavčević
Keyword(s):  
Congruence Lattices
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