The Lattice of Equational Classes of Algebras with One Unary Operation (original) (raw)
On some operations on classes of algebras
Algebra Universalis, 1972
Given any class K of algebras of the same similarity type we let HK and IK denote respectively the classes of all homomorphic and isomorphic images of algebras in K. We also let SK and PK denote respectively the classes of all algebras isomorphic to a subalgebra of a member of K and to a direct product of an arbitrary system of members of K. H, S, P, and I can be conceived of as operations on arbitrary classes of similar algebras and together with all operations constructed from them by composition they are called C-operations. For example, HSP is the operation which at an arbitrary class K takes the value H (S (PK)).
Αγ Algebras - a New Class of Residuated Lattices
2007
An αγ algebra is a residuated lattice satisfying conditions: (C→) : (x→ y)→ (y → x) = y → x and (C∧) : x ∧ y = [x ̄ (x→ y)] ∨ [y ̄ (y → x)], while an α algebra (γ algebra) is a residuated lattice satisfying condition (C→) ((C∧) respectively). Recall that a BL algebra is a bounded residuated lattice satisfying conditions: (prel): (x→ y) ∨ (y → x) = 1 and (div): x ∧ y = x ̄ (x→ y), while a MTL algebra (bounded divisible residuated lattice = bounded commutative Rl-monoid) is a bounded residuated lattice satisfying condition (prel) ((div) respectively). We get: (prel) ⇐⇒ (C→) + (C∨) ⇐⇒ (C∧) + (Cε) and (div) ⇐⇒ (C→)+ (Cδ) ⇐⇒ (C∧) + (Cπ), where (C∨) : x∨y = [(x→ y)→ y]∧ [(y → x)→ x] and the independent conditions (Cδ), (Cε), (Cπ) must be found (open problem). It follows that: (1) bounded αγ algebras are a common generalization of MTL algebras and of bounded divisible residuated lattices; (2) the MTL algebras with condition (DN) (Double Negation): for all x, (x−)− = x, and the bounded αγ a...
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.
Czechoslovak Mathematical Journal, 1985
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On certain lattices associated with generic division algebras
Journal of Group Theory, 2000
Let S n denote the symmetric group on n letters. We consider the S n -lattice A n−1 = {(z 1 , . . . , z n ) ∈ Z n | i z i = 0}, where S n acts on Z n by permuting the coordinates, and its squares A ⊗2 n−1 , Sym 2 A n−1 , and 2 A n−1 . For odd values of n, we show that A ⊗2 n−1 is equivalent to 2 A n−1 in the sense of Colliot-Thélène and Sansuc . Consequently, the rationality problem for generic division algebras amounts to proving stable rationality of the multiplicative invariant field k( 2 A n−1 ) Sn (n odd). Furthermore, confirming a conjecture of Le Bruyn [16], we show that n = 2 and n = 3 are the only cases where A ⊗2 n−1 is equivalent to a permutation S n -lattice. In the course of the proof of this result, we construct subgroups H ≤ S n , for all n that are not prime, so that the multiplicative invariant algebra k[A n−1 ] H has a non-trivial Picard group.
The equational theory of union-free algebras of relations
Algebra Universalis, 1995
We solve a problem of J6nsson by showing that the class Y/of (isomorphs of) algebras of binary relations, under the operations of relative product, conversion, and intersection, and with the identity element as a distinguished constant, is not axiomatizable by a set of equations. We also show that the set of equations valid in ~ is decidable, and in fact the set of equations true in the class of all positive algebras of relations is decidable.
On the construction of free algebras for equational systems
Theoretical Computer Science, 2009
The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applications.