Semisimplicity in Operator Algebras and Subspace Lattices

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 .

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.

Dissertationes Mathematicae, 2001

The plan of the paper is as follows. In Chapter 0 we set up notation and terminology. In Chapter 1 we investigate the properties of consistence, strongness and semimodularity, each of which may be viewed as generalization of modularity. Section 1.1 deals with some conditions characterizing consistence in lower continuous strongly coatomic lattices. Here we prove that a finite lattice is the lattice of closed sets of a closure space with the Steinitz exchange property if and only if it is a consistent lattice. Section 1.2 extends Faigle's concept of strongness from lattices of finite length to arbitrary lattices. Any atomistic lattice is strong whereas the converse does not hold in general. It is shown (Theorem 1.15) that a lower continuous strongly atomic lattice in which each atom has a complement is strong precisely when it is atomistic. For the class of strongly coatomic lower continuous lattices we prove that in semimodular lattices, the concepts of strongness and of consistence are equivalent (Theorem 1.26). Section 1.3 combines semimodularity with the strongness property. In Section 1.4 we characterize atomistic lattices. These characterizations are given in terms of concepts related to pure elements and neat elements. Chapter 2 considers join decompositions in lattices. In Section 2.1, some sufficient conditions are given under which every element of a lattice has a join decomposition (Proposition 2.2). The goal of Section 2.2 is to characterize modularity of lattices in terms of the Kurosh-Ore Replacement Properties (Theorem 2.13). Finally, in Section 2.3, we study lattices with unique irredundant join decompositions. In Chapter 3 Problem IV.15 of Grätzer [1978] is solved. In Section 3.1 we introduce the notion of a c-join in lattices, where c is a distributive element of the lattice. Sections 3.2 and 3.3 present some properties of c-joins and c-decomposition functions. An important role in our investigations is played by the B c-condition defined in Section 3.4. Section 3.5 is devoted to the study of finite c-decompositions of elements in a modular lattice. We give here a generalization of some results of papers of Močulskiȋ [1955, 1961] and Walendziak [1991b]. We find (Theorem 3.25) a common generalization of the Kurosh-Ore Theorem and the Schmidt-Ore Theorem for arbitrary modular lattices, solving a problem of G. Grätzer. In Sections 3.6 and 3.7 we consider infinite c-decompositions. For investigations of such representations, property (B c) does not yield anything. Therefore, we shall use property (B * c) defined in Section 3.6. As a main result of Chapter 3 we give the c-Decomposition Theorem (3.40) which implies (in particular) Crawley's Theorem for direct decompositions (Corollary 3.44) and a generalization of the Kurosh-Ore Theorem to infinite join decompositions (Corollary 3.46). The decomposition theory of Chapters 2 and 3 enables us to devolop a structure theory for algebras. In Chapter 4 we consider weak direct representations of a universal Corollary 1.16. Suppose that a lattice L satisfies the DCC and each atom of L has a complement. Then L is atomistic iff it is strong. Remark 1.17. Since every lattice of finite length satisfies the DCC, this corollary implies the theorem of Stern [1989].

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1990

In memory of Abraham Robinson, on the occasion of his 70th birthday Distributive lattices are locally finite algebras, i.e. their finitely generated sublattices are finite. The general theory of finite distributive lattices is remarkably transparent, due to one or both of the following principles: (i) Any filter (or ideal) is principal. (ii) The lattice is join-generated by its join-irreducible elements, namely by those elements z for which x v y = z implies x = z or y = z. Because of (i), the lattice is isomorphic to its ideal completion, and (ii) is the basis of an elegant coduality between the categories of finite posets and finite distributive lattices. Yet infinite distributive lattices do not, in general, enjoy either of these properties. As a result, there is a plethora of possible completions and the representation theory rests on a rather special category of spaces for which only the Boolean case is well understood. It follows from local finiteness that any distributive lattice D admits hyperfinite extensions D, within any enlargement *D. Now, because the D, are extensions of D, they are flexible enough to contain such classical constructions as the Stone space or the Dedekind-MacNeille completion of D. Because the D, are "finite", only finite lattice theory is available. However, in conjunction with nonstandard methods, such as the transfer principle, this suffices to prove some theorems about infinite lattices. For instance, NACHBIN'S Theorem and STONE'S Prime Filter Theorem as well as several others. While completeness can be expressed in this hyperfinite setting, SIKORSKI'S Extension Theorem faces foundational obstacles (cf. [2]) as already envisaged by W. A. J. LUXEMBURG. There are familiar nonstandard characterizations of general topological notions due to A. ROBINSON. In particular, a subset C of the Stone space S(D) is compact-open exactly if*C is a union of monads. We characterize those lattices for which the hyperfhite version of Ro-BINSON'S condition also suffices to express compact-openness. This class of lattices falls strictly between the Boolean algebras and "T,-lattices" which have been studied before by two of the authors.

Journal of Relation …, 2004

International Journal of Mathematical Archive, 2012

The concept of a dual pseudo-complemented Almost Distributive Lattice is introduced. Necessary and sufficient conditions for an Almost Distributive Lattice to become a dual pseudo-complemented Almost Distributive Lattice are derived. It is proved that a dual pseudo-complemented Almost Distributive Lattice is equationally definable. A one to one correspondence between the set of all dual pseudo-complementations on an ADL and the set of all maximal elements of is obtained. Also proved that the set is a Boolean algebra.

Logical Consequences: Rival Approaches, ed. by John Woods and Bryson Brown, 2001

Mathematical Logic Quarterly, 1999

In this note we introduce and study algebras ( L , V, A, 1, 0,l) of type (2,2,1,1,1) such that ( L , V, A , 0 , l ) is a bounded distributive lattice and -,is an operator that satisfies the conditions -,(a V b ) = -,a A -,b and -0 = 1. We develop the topological duality between these algebras and Priestley spaces with a relation. In addition, we characterize the congruences and the subalgebras of such an algebra. As an application, we will determine the Priestley spaces of quasi-Stone algebras.

Integral Equations and Operator Theory, 2005

We study properties of subspace lattices related to the continuity of the map Lat and the notion of reflexivity. We characterize various "closedness" properties in different ways and give the hierarchy between them. We investigate several properties related to tensor products of subspace lattices and show that the tensor product of the projection lattices of two von Neumann algebras, one of which is injective, is reflexive.

Journal of Pure and Applied Algebra, 2014

In the theory of lattice-ordered groups, there are interesting examples of properties-such as projectability-that are defined in terms of the overall structure of the lattice-ordered group, but are entirely determined by the underlying lattice structure. In this paper, we explore the extent to which projectability is a lattice-theoretic property for more general classes of algebras of logic. For a class of integral residuated lattices that includes Heyting algebras and semilinear residuated lattices, we prove that a member of such is projectable iff the order dual of each subinterval [a, 1] is a Stone lattice. We also show that an integral GMV algebra is projectable iff it can be endowed with a positive Gödel implication. In particular, a ΨMV or an MV algebra is projectable iff it can be endowed with a Gödel implication. Moreover, those projectable involutive residuated lattices that admit a Gödel implication are investigated as a variety in the expanded signature. We establish that this variety is generated by its totally ordered members and is a discriminator variety.