When a lattice homomorphism is a Riesz homomorphism

2006 ◽  
Vol 279 (9-10) ◽  
pp. 1024-1027 ◽  
Author(s):  
Z. Ercan ◽  
A.W. Wickstead
Keyword(s):  
Lattice Homomorphism ◽  
Riesz Homomorphism

scholarly journals Errata: Mal'cev conditions spectra and Kronecker product

1979 ◽  
Vol 28 (4) ◽  
pp. 510-510
Author(s):  
Walter D. Neumann
Keyword(s):  
Kronecker Product ◽  
Lattice Homomorphism ◽  
Finite Algebras

I am grateful to W. Taylor for pointing out that Lemma 2.3 of Neumann (1978), which claimed that a variety and its finitary reduct have the same finite algebras, is incorrect. This weakens several results of the paper: the lattice homomorphism of Lemma 3.1 is not complete, Theorem 4.1 is false, and the Mal'cev conditions claimed to be strong in 6.1, 6.2 and 7.1 are not strong. Taylor (1980) shows one cannot improve on this.

scholarly journals WHEN LATTICE HOMOMORPHISMS OF ARCHIMEDEAN VECTOR LATTICES ARE RIESZ HOMOMORPHISMS

2009 ◽  
Vol 87 (2) ◽  
pp. 263-273 ◽  
Author(s):  
MOHAMED ALI TOUMI
Keyword(s):  
London Math ◽  
Lattice Homomorphism ◽  
Vector Lattices ◽  
Lattice Homomorphisms ◽  
Orthogonal Systems

AbstractLet A, B be Archimedean vector lattices and let (ui)i∈I, (vi)i∈I be maximal orthogonal systems of A and B, respectively. In this paper, we prove that if T is a lattice homomorphism from A into B such that $T\left ( \lambda u_{i}\right ) =\lambda v_{i}$ for each λ∈ℝ+ and i∈I, then T is linear. This generalizes earlier results of Ercan and Wickstead (Math. Nachr279 (9–10) (2006), 1024–1027), Lochan and Strauss (J. London Math. Soc. (2) 25 (1982), 379–384), Mena and Roth (Proc. Amer. Math. Soc.71 (1978), 11–12) and Thanh (Ann. Univ. Sci. Budapest. Eotvos Sect. Math.34 (1992), 167–171).

Congruences on glrac semigroups (I)

2021 ◽  
pp. 2250240
Author(s):  
Haijun Liu ◽  
Xiaojiang Guo
Keyword(s):  
Complete Lattice ◽  
Lattice Homomorphism ◽  
Complete Sublattice ◽  
Restriction Semigroup ◽  
Lattice Of Congruences

The theory of congruences on semigroups is an important part in the theory of semigroups. The aim of this paper is to study [Formula: see text]-congruences on a glrac semigroup. It is proved that the [Formula: see text]-congruences on a glrac semigroup become a complete sublattice of its lattice of congruences. Especially, the structures of left restriction semigroup [Formula: see text]-congruences and the projection-separating [Formula: see text]-congruences on a glrac semigroup are established. Also, we demonstrate that they are both complete sublattice of [Formula: see text]-congruences and consider their relations with respect to complete lattice homomorphism.

scholarly journals On the lattice of congruences on a regular semigroup

1969 ◽  
Vol 1 (2) ◽  
pp. 231-235 ◽  
Author(s):  
T. E. Hall
Keyword(s):  
Regular Semigroup ◽  
Complete Lattice ◽  
Inverse Semigroups ◽  
Natural Homomorphism ◽  
Lattice Homomorphism ◽  
Regular Semigroups ◽  
Lattice Of Congruences

A result of Reilly and Scheiblich for inverse semigroups is proved true also for regular semigroups. For any regular semigroup S the relation θ is defined on the lattice, Λ(S), of congruences on S by: (ρ, τ) ∈ θ if ρ and τ induce the same partition of the idempotents of S. Then θ is a congruence on Λ(S), Λ(S)/θ is complete and the natural homomorphism of Λ(S) onto Λ(S)/θ is a complete lattice homomorphism.

scholarly journals Dualities for some De Morgan algebras with operators and Lukasiewicz algebras

1983 ◽  
Vol 34 (3) ◽  
pp. 377-393 ◽  
Author(s):  
Roberto Cignoli ◽  
Marta S. De Gallego
Keyword(s):  
Lattice Homomorphism ◽  
De Morgan Algebra ◽  
De Morgan Algebras

AbstractAlgebras (A, ∧, ∨, ~, γ, 0, 1) of type (2,2,1,1,0,0) such that (A, ∧, ∨, ~, γ 0, 1) is a De Morgan algebra and γ is a lattice homomorphism from A into its center that satisfies one of the conditions (i) a ≤ γa or (ii) a ≤ ~ a ∧ γa are considered. The dual categories and the lattice of their subvarieties are determined, and applications to Lukasiewicz algebras are given.

Homomorphisms of Distributive Lattices as Restrictions of Congruences

1986 ◽  
Vol 38 (5) ◽  
pp. 1122-1134 ◽  
Author(s):  
George Grätzer ◽  
Harry Lakser
Keyword(s):  
Distributive Lattice ◽  
Congruence Lattice ◽  
Classical Result ◽  
Finite Lattice ◽  
Distributive Lattices ◽  
Lattice Homomorphism ◽  
Finite Distributive Lattice ◽  
Full Generality ◽  
Ideal Lattices

Given a lattice L and a convex sublattice K of L, it is well-known that the map Con L → Con K from the congruence lattice of L to that of K determined by restriction is a lattice homomorphism preserving 0 and 1. It is a classical result (first discovered by R. P. Dilworth, unpublished, then by G. Grätzer and E. T. Schmidt [2], see also [1], Theorem II.3.17, p. 81) that any finite distributive lattice is isomorphic to the congruence lattice of some finite lattice. Although it has been conjectured that any algebraic distributive lattice is the congruence lattice of some lattice, this has not yet been proved in its full generality. The best result is in [4]. The conjecture is true for ideal lattices of lattices with 0; see also [3].

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