Lattice of quasiorders of monounary algebras (original) (raw)

Algebra and Discrete Mathematics Construction of a complementary quasiorder *

2018

For a monounary algebra A = (A, f ) we study the lattice Quord A of all quasiorders of A, i.e., of all reflexive and transitive relations compatible with f . Monounary algebras (A, f ) whose lattices of quasiorders are complemented were characterized in 2011 as follows: ( * ) f (x) is a cyclic element for all x ∈ A, and all cycles have the same square-free number n of elements. Sufficiency of the condition ( * ) was proved by means of transfinite induction. Now we will describe a construction of a complement to a given quasiorder of (A, f ) satisfying ( * ).

Congruence lattices of connected monounary algebras

Algebra Universalis, 2020

The system of all congruences of an algebra (A, F) forms a lattice, denoted {{\\,\\mathrm{Con}\\,}}(A, F).Further,thesystemofallcongruencelatticesofallalgebraswiththebasesetAformsalattice. Further, the system of all congruence lattices of all algebras with the base set A forms a lattice.Further,thesystemofallcongruencelatticesofallalgebraswiththebasesetAformsalattice\\mathcal {E}_A.Wedealwithmeet−irreducibilityin. We deal with meet-irreducibility in.Wedealwithmeet−irreducibilityin\\mathcal {E}_AforagivenfinitesetA.Allmeet−irreducibleelementsoffor a given finite set A. All meet-irreducible elements offoragivenfinitesetA.Allmeet−irreducibleelementsof\\mathcal {E}_Aarecongruencelatticesofmonounaryalgebras.Sometypesofmeet−irreduciblecongruencelatticeswerealreadydescribed.Inthecasewhenamonounaryalgebra(A,f)isconnected,weprovenecessaryandsufficientconditionunderwhichare congruence lattices of monounary algebras. Some types of meet-irreducible congruence lattices were already described. In the case when a monounary algebra (A, f) is connected, we prove necessary and sufficient condition under whicharecongruencelatticesofmonounaryalgebras.Sometypesofmeet−irreduciblecongruencelatticeswerealreadydescribed.Inthecasewhenamonounaryalgebra(A,f)isconnected,weprovenecessaryandsufficientconditionunderwhich{{\\,\\mathrm{Con}\\,}}(A, f)isisis\\wedge -irreducible.

Some properties of retract lattices of monounary algebras

Mathematica Slovaca, 2012

Necessary and sufficient conditions for a connected monounary algebra (A, f), under which the lattice R ∅(A, f) of all retracts of (A, f) (together with ∅) is algebraic, are proved. Simultaneously, all connected monounary algebras in which each retract is a union of completely join-irreducible elements of R ∅(A, f) are characterized. Further, there are described all connected monounary algebras (A, f) such that the lattice R ∅(A, f) is complemented. In this case R ∅(A, f) forms a boolean lattice.

Construction of a complementary quasiorder

Algebra and discrete mathematics, 2018

For a monounary algebra A = (A, f) we study the lattice Quord A of all quasiorders of A, i.e., of all reflexive and transitive relations compatible with f. Monounary algebras (A, f) whose lattices of quasiorders are complemented were characterized in 2011 as follows: (*) f (x) is a cyclic element for all x ∈ A, and all cycles have the same square-free number n of elements. Sufficiency of the condition (*) was proved by means of transfinite induction. Now we will describe a construction of a complement to a given quasiorder of (A, f) satisfying (*).

Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond

2020

Noncommutative Lattices Skew Lattices, Skew Boolean Algebras and Beyond famnit lectures ■ famnitova predavanja ■ 4 Jonathan E. Leech Proof. Given x∧z = y∧z and x∨z = y∨z, then x = x ∨ (x ∧ z) = x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) = (x ∨ y) ∧ (y ∨ z) = (x ∨ z) ∧ y ≤ y and similarly, y ≤ x, so that x = y. Conversely, neither M 3 nor N 5 can be subalgebras of a cancellative slew lattice. £ A lattice (L; ∨, ∧) is complete if every subset X of L has a supremum (an element u ≥ x for all x in X, with u being the least such element in L) denoted by sup(X) and an infimum (an element v ≤ x for all x in X, with v being the greatest such element in L) denoted by inf(X). In particular, a complete lattice has a greatest element 1 and a least element 0. Conversely, a lattice with both least and greatest elements 0 and 1 is complete if all subsets have suprema, or equivalently, if all subsets have infima. Finally, in any complete lattice, we let 0 = sup(∅) and 1 = inf(∅). Lattices and universal algebra An algebra is any system, A = (A: f 1 , f 2 , …, f r), where A is a set and each f i is an n i-ary operation on A. If B ⊆ A is such that for all i ≤ r, f i (b 1 , b 2 , …, b n i) ∈ B for all b 1 , …, b n i in B, then the system B = (B: f 1 ʹ, f 2 ʹ, …, f r ʹ) where f i ʹ= f i ⎢ B n i is a subalgebra of A. (When confusion occurs, subalgebras may be indicated by their underlying sets.) Under inclusion, ⊆, the subalgebras of an algebra A form a complete lattice Sub(A) with greatest element A, least element the smallest subalgebra containing ∅ and meets given by intersection. If none of the operations are nullary, then the least subalgebra is the empty subalgebra, ∅. If there are no operations, then Sub(A) is the lattice 2 A. Recall that a congruence on A = (A: f 1 , f 2 , …, f r) is an equivalence relation θ on A such that given i ≤ r with a 1 θb 1 , a 2 θb 2 , …, a n i θ b n i in A, then f i (a 1 , a 2 , …, a n i) θ f i (b 1 , b 2 , …, b n i). Under inclusion, ⊆, the congruences on A form a complete lattice Con(A). Its greatest element is the universal relation ∇ = A×A relating all elements in A. Its least element is the identity relation Δ. Suprema and infima in Con(A) are calculated as in the lattice Equ(A) of all equivalences on A. In particular, infima in Con(A) are given by intersection. £ Recall that an element c in a lattice (L; ∨, ∧) is compact if for any subset X of L, c ≤ supX implies that c ≤ supY for some finite subset Y of X. (Every cover can be reduced to a finite cover.) An algebraic lattice is a complete lattice for which every element is a supremum of compact elements. The proof of the following result is easily accessible in the literature Theorem 1.1.4. Given an algebra A = (A: f 1 , f 2 , …, f r), both Sub(A) and Con(A) are algebraic lattices. I: Preliminaries Of particular interest is the next result. It's proof may be obtained in any standard text on lattice theory. Theorem 1.1.5. Congruence lattices of lattices are distributive. £ A subset U of a poset (L; ≥) is directed upward if given any two elements x, y in U, a third element z exists in U such that x, y ≤ z. The proof of the next result is also easily accessible. Theorem 1.1.6. Given an algebraic lattice (L; ∨, ∧), a ∧ sup(U) = sup{a∧x ⎢x ∈ U} holds if U is directed upward. This equality holds unconditionally when (L; ∨, ∧) is also distributive. Recall that two algebras A = (A; f 1 , f 2 , …, f r) and B = (B; g 1 , g 2 , …, g s) have the same type if r = s and for all i ≤ r, both f i and g i have the same number of variables, that is, both are say n i-ary operations. Recall also that a class V of algebras of the same type is a variety if it is closed under direct products, subalgebras and homomorphic images. A classic result of Birkhoff is as follows: Theorem 1.1.7. Among algebras of the same type, each variety is determined by the set of all identities satisfied by all algebras in that variety. That is, all varieties are equationally determined in the class of all algebras of the same type. £ Proof. That χ is a homomorphism follows easily from the associative, commutative and distributive laws. By cancellation, χ is one-to-one. Upon composing with either coordinate projections, it clearly it mapped onto each factor. £ Corollary 1.1.10. Every nontrivial distributive lattice is a subdirect product of C 1. £ We return to the variety of all lattices. On any lattice, consider the polynomial M(x, y, z) = (x∨y) ∧ (x∨z) ∧ (y∨z) that was implicit in the proof of Theorem 1.5. M satisfies the identities M(x, x, y) = M(x, y, x) = M(y, x, x) = x. Given an algebra A = (A; f 1 , …, f r) on which a ternary operation M(x, y, z) satisfying these identities is polynomial-defined using the operations of A, then Con(A) is distributive. In general, if a ternary function M can be defined from the functions symbols of a variety V such that M satisfied these identities on all algebras in V, then the congruence lattices of all algebras in that variety are distributive and V is said to be congruence distributive. Boolean lattices and Boolean algebras Given a lattice (L; ∧, ∨) with maximal and minimal elements 1 and 0, elements x and xʹ are complements in L if x∨xʹ = 1 and x∧xʹ = 0. If L is distributive, then the complement xʹ of any element x is unique. Indeed, let xʺ be a second complement of x. Then xʺ = xʺ ∧ 1 = xʺ ∧ (x ∨ xʹ) = (xʺ ∧ x) ∨ (xʺ ∧ xʹ) = 0 ∨ (xʺ ∧ xʹ) = xʺ ∧ xʹ. Similarly, xʹ = xʹ ∧ xʺ and xʹ = xʺ follows. Clearly 0 and 1 are mutual complements. Recall that Boolean lattice is a distributive lattice with maximal and minimal elements 1 and 0, (L; ∧, ∨, 1, 0), such that every x in L has a (necessarily unique) complement xʹ in L. If the operation ʹ is built into the signature, then (L; ∧, ∨, ʹ, 1, 0) is a Boolean algebra. Boolean algebras are characterized by the identities for a distributive lattice augmented by the identities for maximal and minimal elements and the identities for complementation. They also satisfy the DeMorgan identities: (x ∨ y)ʹ = xʹ ∧ yʹ and (x ∧ y)ʹ = xʹ ∨ yʹ. Given a Boolean algebra, the difference (or relative complement) of elements x and y is defined by x \ y = x ∧ yʹ. This operation satisfies the relative DeMorgan identities: x \ (y ∨ z) = (x\y) ∧ (x\z) and x \ (y ∧ z) = (x\y) ∨ (x\z). More generally, given any distributive lattice with a maximum 1 and minimum 0, if x and y have complements, then so do x ∨ y and x ∧ y with (x ∨ y)ʹ = xʹ ∧ yʹ and (x ∧ y)ʹ = xʹ ∨ yʹ.

On the semidistributivity of elements in weak congruence lattices of algebras and groups

Algebra universalis, 2008

Weak congruence lattices and semidistributive congruence lattices are both recent topics in Universal Algebra. This motivates the main result of the present paper, which asserts that a finite group G is a Dedekind group if and only if the diagonal relation is a join-semidistributive element in the lattice of weak congruences of G. A variant in terms of subgroups rather than weak congruences is also given. It is pointed out that no similar result is valid for rings. An open problem and some results on the join-semidistributivity of weak congruence lattices are also included.

Subalgebras, direct products and associated lattices of MV-algebras

Glasgow Mathematical Journal, 1992

0. Introduction. MV-algebras were introduced by C. C. Chang in 1958 in order to provide an algebraic proof for the completeness theorem of the Lukasiewicz infinite valued propositional logic. In recent years the scope of applications of MV-algebras has been extended to lattice-ordered abelian groups, AF C*-algebras [10] and fuzzy set theory In [1] Belluce defined a functor y from MV-algebras to bounded distributive lattices; this functor was used in proving a representation theorem and was also used to show that the prime ideal space of an MV-algebra is homeomorphic to the prime ideal space of some bounded distributive lattice (both spaces endowed with the Stone topology). The problem of what the range of y is arises naturally. This question bears a relation to the question as to whether there is an "MV-space" in the same manner as there are Boolean spaces for Boolean algebras. Some "MV-spaces" are considered by N. G. Martinez .

The congruence lattice of implication algebras

The variety of implicat.ion algebras is a minimal quasi variety. It is 3-filtral but not 2-filtral. An implication algebra A is tolerance-trivial iff (A,:S) is a lattice, where the partial ordering I':S" is defined as follows: a :S b {:} 3x E A such that b = x. a.

Lattice-Based Relation Algebras II

Lecture Notes in Computer Science, 2006

We present classes of algebras which may be viewed as weak relation algebras, where a Boolean part is replaced by a not necessarily distributive lattice. For each of the classes considered in the paper we prove a relational representation theorem.