Lattice-ordered triangular matrix algebras

Mathematica Slovaca, 2016

A mutual relationship between MV-algebras and coupled semirings as established by L. P. Belluce, A. Di Nola, A. R. Ferraioli and B. Gerla is extended to lattice effect algebras and so-called characterizing triples. We show that this correspondence is in fact one-to-one and hence every lattice effect algebra can be considered as an ordered triple consisting of two semiring-like structures and an antitone involution which is an isomorphism between these structures.

International Journal of Theoretical Physics, 2016

Since every lattice effect algebra decomposes into blocks which are MV-algebras and since every MV-algebra can be represented by a certain semiring with an antitone involution as shown by Belluce, Di Nola and Ferraioli, the natural question arises if a lattice effect algebra can also be represented by means of a semiring-like structure. This question is answered in the present paper by establishing a one-to-one correspondence between lattice effect algebras and certain right near semirings with an antitone involution. Keywords Effect algebra • Lattice effect algebra • Right near semiring • Antitone involution • Effect near semiring Effect algebras were introduced by Foulis and Bennett [8] in order to axiomatize unsharp logics of quantum mechanics. Although the definition of an effect algebra looks elementary, these algebras have several very surprising properties. Concerning these properties the reader is referred to the monograph [7] by Dvurečenskij and Pulmannová. In particular, every effect algebra induces a natural partial order relation and thus can be considered as a bounded poset. If this poset is a lattice, the effect algebra is called a lattice effect algebra.

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.

Annals of Pure and Applied Logic, 2009

It is shown that the Boolean center of complemented elements in a bounded integral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FL w -algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded n-potent GBL-algebras are represented as Esakia products of simple n-potent MV-algebras.

Outstanding Contributions to Logic, 2014

A variety V of lattice-ordered algebras is said to be semi-linear in case it is generated by its totally ordered members (in more traditional algebraic parlance, the term 'representable' is often used in place of 'semi-linear'). Due to the congruence distributivity of V , V is semi-linear if and only if its subdirectly irreducible members are totally ordered (Burris and Sankappanavar, 1981, Theorem 6.8, p. 165). Needless to say, semi-linearity is a welcome property insofar as it often makes a class of algebras very tractable for computation and proof purposes. Many well-understood varieties in algebraic logic are known to be semi-linear: examples include Abelian-groups and varieties arising from many-valued logic (such as MTL algebras and thus, in particular, BL algebras, MV algebras or Gödel algebras: Cintula et al. 2011). Petr Hájek, besides giving fundamental contributions to the investigation of many such classes of algebras, has repeatedly underscored the central role played by semilinearity in fuzzy logic: Among the logics of residuated lattices, fuzzy logics [...] are distinguished by the property of semilinearity, i.e., completeness w.r.t. a class of linearly ordered residuated lattices. The

Illinois journal of mathematics

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 .

In 1992 Alexandru Nica published a highly influential paper examining the C*-algebras generated by a semigroup he calls a quasi-lattice ordered group. Of particular interest to Nica were the C*-algebra generated by the Toeplitz representation of a quasi-lattice order, and a universally defined C*-algebra he calls C*(G, P). Due to the concreteness of the first C*-algebra and the universal properties of the second, Nica was interested in finding sufficient conditions to determine when these two algebras were isomorphic. He called quasi-lattice orders that satisfied this isomorphism condition amenable. Finding techniques that establish amenability is now a topic at the cutting edge of research in the study of C*-algebras of semigroups. In this dissertation, we follow the outline set out by Nica's paper to study the C*-algebras generated by weakly quasi-lattice ordered groups. Most of the proofs throughout this dissertation required a substantial amount of original work to fill in details omitted by Nica, as he often only offered a proof outline.

Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, 2019

In this paper, we introduce the algebra M n n (L) of square matrices over residuated lattice L. The operations are induced by the corresponding operations of L. It is shown that the de…ned algebra behaves like a residuated lattice, but there are some slight di¤erences. The properties of this algebra with respect to special residuated lattices are investigated. The notions of …lter and ideal together with their roles are speci…ed.

Fuzzy Sets and Systems

We introduce the concept of a quasiresiduated lattice and prove that every lattice effect algebra can be organized into a commutative quasiresiduated lattice with divisibility. Also conversely, every such lattice can be converted into a lattice effect algebra and every lattice effect algebra can be reconstructed form its assigned quasiresiduated lattice. We apply this method also for lattice pseudoeffect algebras introduced recently by Dvurečenskij and Vetterlein. We show that every good lattice pseudoeffect algebra can be organized into a (possibly non-commutative) quasiresiduated lattice with divisibility and conversely, every such lattice can be converted into a lattice pseudoeffect algebra. Moreover, also a good lattice pseudoeffect algebra can be reconstructed from the assigned quasiresiduated lattice.