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Bezhanishvili, Guram, and Patrick J. Morandi. "Profinite Heyting Algebras and Profinite Completions of Heyting Algebras." gmj 16, no. 1 (March 2009): 29–47. http://dx.doi.org/10.1515/gmj.2009.29.
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Abstract This paper surveys recent developments in the theory of profinite Heyting algebras (resp. bounded distributive lattices, Boolean algebras) and profinite completions of Heyting algebras (resp. bounded distributive lattices, Boolean algebras). The new contributions include a necessary and sufficient condition for a profinite Heyting algebra (resp. bounded distributive lattice) to be isomorphic to the profinite completion of a Heyting algebra (resp. bounded distributive lattice). This results in simple examples of profinite bounded distributive lattices that are not isomorphic to the profinite completion of any bounded distributive lattice. We also show that each profinite Boolean algebra is isomorphic to the profinite completion of some Boolean algebra. It is still an open question whether each profinite Heyting algebra is isomorphic to the profinite completion of some Heyting algebra.
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Ali M. Ali Rushdi, Ali M. Ali Rushdi. "Satisfiability in Big Boolean Algebras via Boolean-Equation Solving." journal of King Abdulaziz University Engineering Sciences 28, no. 1 (January 2, 2017): 3–18. http://dx.doi.org/10.4197/eng.28-1.1.
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This paper studies Satisfiability (SAT) in finite atomic Boolean algebras larger than the two-valued one B2, which are named big Boolean algebras. Unlike the formula ݃(ࢄ (in the SAT problem over B2, which is either satisfiable or unsatisfiable, this formula for the SAT problem over a big Boolean algebra could be unconditionally satisfiable, conditionally satisfiable, or unsatisfiable depending on the nature of the consistency condition of the Boolean equation {݃(ࢄ = (1}, since this condition could be an identity, a genuine equation, or a contradiction. The paper handles this latter SAT problem by using a conventional method and a novel one for deriving parametric general solutions, and subsequently utilizing expansion trees for generating all particular solutions of the aforementioned Boolean equation. Each of these two methods could be cast in pure algebraic form, but becomes much easier to visualize and comprehend when presented via the natural map of a big Boolean algebra, which (for historical reasons) is called the variable-entered Karnaugh map (VEKM). In the classical method, the number of parameters used is minimized and compact solutions are obtained. However, the parameters belong to the underlying big Boolean algebra. By contrast, the novel method does not attempt to minimize the number of parameters used, as it uses independent parameters belonging to the two-valued Boolean algebra B2 for each asserted atom in the Boole-Shannon expansion of the formula ݃(ࢄ .(Though the method produces non-compact expressions, it is much quicker in generating particular solutions. The two methods are demonstrated via two detailed examples.
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Ehrenfeucht, Andrzej, and Marek W. Zawadowski. "A Representation of Partial Boolean Algebras." Fundamenta Informaticae 16, no. 3-4 (May 1, 1992): 349–53. http://dx.doi.org/10.3233/fi-1992-163-408.
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Knight, Julia F., and Michael Stob. "Computable Boolean algebras." Journal of Symbolic Logic 65, no. 4 (December 2000): 1605–23. http://dx.doi.org/10.2307/2695066.
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Feiner [F] showed that a Boolean algebra need not have a computable copy (see also [T2]). Downey and Jockusch [D-J] showed that every low Boolean algebra does have a computable copy. Thurber [T3], showed that every low2 Boolean algebra has a computable copy. Here we show that every Boolean algebra which is low3, or even low4, has a computable copy.The results of [D-J] and [T3] were obtained by passing to linear orderings. In [D-J], there is an embedding theorem saying that any linear ordering which is with the successor relation as an added predicate can be embedded in a slightly larger linear ordering which is computable. An isomorphism theorem of Remmel [R] is used to show that the interval algebras of the two linear orderings are isomorphic (except in a trivial case). In [T3], there is an embedding theorem saying that any linear ordering which is with certain added predicates can be embedded in one which is with successor. Again the isomorphism theorem of Remmel is used to show that the interval algebras are isomorphic (except in a trivial case).Here, instead of passing to linear orderings, we work directly with Boolean algebras. We begin with a review of the known results. We re-formulate the embedding theorems of Downey-Jockusch and Thurber in terms of Boolean algebras. We extract from Remmel's isomorphism theorem some information on complexity. In this way, we show that a low Boolean algebra is isomorphic to a computable one by an isomorphism which is , at worst, and the same is true for a low2 Boolean algebra.
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BEZHANISHVILI, NICK, and WESLEY H. HOLLIDAY. "CHOICE-FREE STONE DUALITY." Journal of Symbolic Logic 85, no. 1 (August 29, 2019): 109–48. http://dx.doi.org/10.1017/jsl.2019.11.
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AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.
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Rushdi, Ali Muhammad Ali, and Waleed Ahmad. "Digital Circuit Design Utilizing Equation Solving over ‘Big’ Boolean Algebras." International Journal of Mathematical, Engineering and Management Sciences 3, no. 4 (December 1, 2018): 404–28. http://dx.doi.org/10.33889/ijmems.2018.3.4-029.
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A task frequently encountered in digital circuit design is the solution of a two-valued Boolean equation of the form h(X,Y,Z)=1, where h: B_2^(k+m+n)→ B_2 and X,Y, and Z are binary vectors of lengths k, m, and n, representing inputs, intermediary values, and outputs, respectively. The resultant of the suppression of the variables Y from this equation could be written in the form g(X,Z)=1 where g: B_2^(k+n)→ B_2. Typically, one needs to solve for Z in terms of X, and hence it is unavoidable to resort to ‘big’ Boolean algebras which are finite (atomic) Boolean algebras larger than the two-valued Boolean algebra. This is done by reinterpreting the aforementioned g(X,Z) as g(Z): B_(2^K)^n→ B_(2^K ), where B_(2^K ) is the free Boolean algebra FB(X_1,X_2…….X_k ), which has K= 2^k atoms, and 2^K elemnets. This paper describes how to unify many digital specifications into a single Boolean equation, suppress unwanted intermediary variables Y, and solve the equation g(Z)=1 for outputs Z (in terms of inputs X) in the absence of any information about Y. The paper uses a novel method for obtaining the parametric general solutions of the ‘big’ Boolean equation g(Z)=1. The parameters used do not belong to B_(2^K ) but they belong to the two-valued Boolean algebra B_2, also known as the switching algebra or propositional algebra. To achieve this, we have to use distinct independent parameters for each asserted atom in the Boole-Shannon expansion of g(Z). The concepts and methods introduced herein are demonsrated via several detailed examples, which cover the most prominent type among basic problems of digital circuit design.
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Bonnet, Robert, and Matatyahu Rubin. "Elementary embedding between countable Boolean algebras." Journal of Symbolic Logic 56, no. 4 (December 1991): 1212–29. http://dx.doi.org/10.2307/2275469.
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AbstractFor a complete theory of Boolean algebras T, let MT denote the class of countable models of T. For B1, B2 ∈ MT, let B1 ≤ B2 mean that B2 is elementarily embeddable in B2. Theorem 1. For every complete theory of Boolean algebras T, if T ≠ Tω, then ‹MT, ≤› is well-quasi-ordered. ∎ We define Tω. For a Boolean algebra B, let I(B) be the ideal of all elements of the form a + s such that B ↾ a is an atomic Boolean algebra and B ↾ s is an atomless Boolean algebra. The Tarski derivative of B is defined as follows: B(0) = B and B(n + 1) = B(n)/I(B(n)). Define Tω to be the theory of all Boolean algebras such that for every n ∈ ω, B(n) ≠ {0}. By Tarski [1949], Tω is complete. Recall that ‹A, < › is partial well-quasi-ordering, it is a partial quasi-ordering and for every {ai, ⃒ i ∈ ω} ⊆ A, there are i < j < ω such that ai ≤ aj. Theorem 2. contains a subset M such that the partial orderings ‹M, ≤ ↾ M› and are isomorphic. ∎ Let M′0 denote the class of all countable Boolean algebras. For B1, B2 ∈ M′0, let B1 ≤′ B2 mean that B1 is embeddable in B2. Remark. ‹M′0, ≤′› is well-quasi-ordered. ∎ This follows from Laver's theorem [1971] that the class of countable linear orderings with the embeddability relation is well-quasi-ordered and the fact that every countable Boolean algebra is isomorphic to a Boolean algebra of a linear ordering.
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Paad, Akbar. "Ideals in bounded equality algebras." Filomat 33, no. 7 (2019): 2113–23. http://dx.doi.org/10.2298/fil1907113p.
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In this paper, the concept of ideal in bounded equality algebras is introduced. With respect to this concepts, some related results are given. In particular, we prove that there is an one-to-one corresponding between congruence relation on an involutive equality algebra and the set of ideals on it. Also, we prove the first isomorphism theorem on equality algebras. Moreover, the notions of prime and Boolean ideals in equality algebras are introduced. Finally, we prove that ideal I of involutive prelinear equality algebra E is a Boolean ideal if and only if E/I is a Boolean algebra.
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Rao, G. C., and P. Sundarayya. "Boolean Algebra of C-Algebras." ITB Journal of Sciences 44, no. 3 (2012): 204–16. http://dx.doi.org/10.5614/itbj.sci.2012.44.3.1.
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Džamonja, Mirna, and Grzegorz Plebanek. "Strictly positive measures on Boolean algebras." Journal of Symbolic Logic 73, no. 4 (December 2008): 1416–32. http://dx.doi.org/10.2178/jsl/1230396929.
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AbstractWe investigate strictly positive finitely additive measures on Boolean algebras and strictly positive Radon measures on compact zerodimensional spaces. The motivation is to find a combinatorial characterisation of Boolean algebras which carry a strictly positive finitely additive finite measure with some additional properties, such as separability or nonatomicity. A possible consistent characterisation for an algebra to carry a separable strictly positive measure was suggested by Talagrand in 1980, which is that the Stone space K of the algebra satisfies that its space M(K) of measures is weakly separable, equivalently that C(K) embeds into l∞. We show that there is a ZFC example of a Boolean algebra (so of a compact space) which satisfies this condition and does not support a separable strictly positive measure. However, we use this property as a tool in a proof which shows that under MA + ¬ CH every atomless ccc Boolean algebra of size < c carries a nonatomic strictly positive measure. Examples are given to show that this result does not hold in ZFC. Finally, we obtain a characterisation of Boolean algebras that carry a strictly positive nonatomic measure in terms of a chain condition, and we draw the conclusion that under MA + ¬ CH every atomless ccc Boolean algebra satisfies this stronger chain condition.
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Fuchino, Sakaé. "Some remarks on openly generated Boolean algebras." Journal of Symbolic Logic 59, no. 1 (March 1994): 302–10. http://dx.doi.org/10.2307/2275267.
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AbstractA Boolean algebra B is said to be openly generated if {A : A ≤rcB, ∣A∣ = ℵ0} includes a club subset of . We show:(V = L). For any cardinal κ there exists an L∞κ-free Boolean algebra which is not openly generated (Proposition 4.1).(MA+(σ-closed)). Every -free Boolean algebra is openly generated (Theorem 4.2).The last assertion follows from a characterization of openly generated Boolean algebras under MA+(σ-closed) (Theorem 3.1). Using this characterization we also prove the independence of problem 7 in Ščepin [15] (Proposition 4.3 and Theorem 4.4).
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Koppelberg, Sabine, and Saharon Shelah. "Densities of Ultraproducts of Boolean Algebras." Canadian Journal of Mathematics 47, no. 1 (February 1, 1995): 132–45. http://dx.doi.org/10.4153/cjm-1995-007-0.
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AbstractWe answer three problems by J. D. Monk on cardinal invariants of Boolean algebras. Two of these are whether taking the algebraic density πA resp. the topological density cL4 of a Boolean algebra A commutes with formation of ultraproducts; the third one compares the number of endomorphisms and of ideals of a Boolean algebra.
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Ricker, Werner. "Uniformly Closed Algebras Generated by Boolean Algebras of Projections in Locally Convex Spaces." Canadian Journal of Mathematics 39, no. 5 (October 1, 1987): 1123–46. http://dx.doi.org/10.4153/cjm-1987-057-5.
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The theory of operator algebras in Banach spaces generated by Boolean algebras of projections is by now well known. It is systematically exposed in the penetrating studies of W. Bade, [1], [2] and [6, Chapter XVII]. Many of these results, a priori independent on normability of the underlying space, have recently been extended to the setting of locally convex spaces; see [3], [4], [5], [11] and [15], for example.However, one of Bade's fundamental results, stating that the closed algebra generated by a complete Boolean algebra in the uniform operator topology is the same as the closed algebra that it generates in the weak operator topology, has remained remarkably resistant in attempts to extend it to locally convex spaces. Recently however, a class of Boolean algebras in non-normable spaces, called boundedly σ-complete Boolean algebras, was exhibited in which the analogue of Bade's result is valid, [14; Theorem 5.3].
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Balcar, Bohuslav, and Thomas Jech. "Weak Distributivity, A Problem of Von Neumann and the Mystery of Measurability." Bulletin of Symbolic Logic 12, no. 2 (June 2006): 241–66. http://dx.doi.org/10.2178/bsl/1146620061.
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This article investigates the weak distributivity of Boolean σ-algebras satisfying the countable chain condition. It addresses primarily the question when such algebras carry a σ-additive measure. We use as a starting point the problem of John von Neumann stated in 1937 in the Scottish Book. He asked if the countable chain condition and weak distributivity are sufficient for the existence of such a measure.Subsequent research has shown that the problem has two aspects: one set theoretic and one combinatorial. Recent results provide a complete solution of both the set theoretic and the combinatorial problems. We shall survey the history of von Neumann's Problem and outline the solution of the set theoretic problem. The technique that we describe owes much to the early work of Dorothy Maharam to whom we dedicate this article.§1. Complete Boolean algebras and weak distributivity. ABoolean algebrais a setBwith Boolean operationsa˅b(join),a˄b(meet) and −a(complement), partial orderinga≤bdefined bya˄b=aand the smallest and greatest element,0and1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty setS, under operationsa∪b,a∩b,S−a, ordered by inclusion, with0= ∅ and1=S.Complete Boolean algebras and weak distributivity.A Boolean algebrais a setBwith Boolean operationsa˅b(join),a˄b(meet) and -a(complement), partial orderinga≤bdefined bya˄b=aand the smallest and greatest element.0and1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty setS, under operationsa∪b,a∩b,S-a, ordered by inclusion, with0= ϕ and1=S.
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De Villiers, Michael D. "Teaching Modeling and Axiomatization with Boolean Algebra." Mathematics Teacher 80, no. 7 (October 1987): 528–32. http://dx.doi.org/10.5951/mt.80.7.0528.
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Traditionally, Boolean algebra is largely taught in connection with computer programming courses, logic, or set theory. Since Boolean algebra arose from George Boole's application of algebraic principles to the study of logic in 1854, this approach would seem natural.
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Andréka, Hajnal, Steven Givant, and István Németi. "Perfect extensions and derived algebras." Journal of Symbolic Logic 60, no. 3 (September 1995): 775–96. http://dx.doi.org/10.2307/2275756.
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Jónsson and Tarski [1951] introduced the notion of a Boolean algebra with (additive) operators (for short, a Bo). They showed that every Bo can be extended to a complete and atomic Bo satisfying certain additional conditions, and that any two complete, atomic extensions of satisfying these conditions are isomorphic over . Henkin [1970] extended these results to Boolean algebras with generalized (i.e., weakly additive) operators. The particular complete, atomic extension of studied by Jónsson and Tarski is called the perfect extension of , and is denoted by +. It is very useful in algebraic investigations of classes of algebras that are associated with logics.Interesting examples of Bos abound in algebraic logic, and include relation algebras, cylindric algebras, and polyadic and quasi-polyadic algebras (with or without equality). Moreover, there are several important constructions that, when applied to certain Bos, lead to other, derived Bos. Obvious examples include the formation of subalgebras, homomorphic images, relativizations, and direct products. Other examples include the Boolean algebra of ideal elements of a Bo, the neat β;-reduct of an α-dimensional cylindric algebra (β; < α), and the relation algebraic reduct of a cylindric algebra (of dimension at least 3). It is natural to ask about the relationship between the perfect extension of a Bo and the perfect extension of one of its derived algebras ′: Is the perfect extension of the derived algebra just the derived algebra of the perfect extension? In symbols, is (′)+ = (+)′? For example, is the perfect extension of a subalgebra, homomorphic image, relativization, or direct product, just the corresponding subalgebra, homomorphic image, relativization, or direct product of the perfect extension (up to isomorphisms)? Is the perfect extension of the Boolean algebra of ideal elements, or the neat reduct of a cylindric algebra, or the relation algebraic reduct of a cylindric algebra just the Boolean algebra of ideal elements, or the neat β;-reduct, or the relation algebraic reduct, of the perfect extension? We shall prove a general result in this direction; namely, if the derived algebra is constructed as the range of a relatively multiplicative operator, then the answer to our question is “yes”. We shall also give examples to show that in “infinitary” constructions, our question can have a spectacularly negative answer.
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Reddy, J. K. "Boolean Algebra." IETE Journal of Education 28, no. 4 (October 1987): 153–58. http://dx.doi.org/10.1080/09747338.1987.11436173.
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Brinn, L. W. "Boolean algebra." International Journal of Mathematical Education in Science and Technology 20, no. 6 (November 1989): 799–807. http://dx.doi.org/10.1080/0020739890200602.
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Paad, Akbar. "Integral prefilters and integral Eq-algebras." Mathematica Slovaca 72, no. 2 (March 28, 2022): 287–300. http://dx.doi.org/10.1515/ms-2022-0019.
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Abstract In this paper, the notion of some new classes of EQ-algebras which are called integral, O-local, and locally finite EQ-algebras are introduced, and relationships between locally finite EQ-algebras and maximal prefilters are described. Also, the notion of integral prefilters are introduced and several characteristics of them are presented. Moreover, relation among integral prefilters and some type of other prefilter such as positive implicative, maximal, prime, and fantastic prefilters in EQ-algebras are studied. Finally, Boolean prefilters in bounded lattice EQ-algebras are introduced, and it is proved that a filter of a prelinear involutive EQ-algebra L is a Boolean filter if and only if L/F is a Boolean algebra.
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Pushpalatha, K., and V. M.L.Hima Bindu. "A Note on Boolean Like Algebras." International Journal of Engineering & Technology 7, no. 4.10 (October 2, 2018): 1015. http://dx.doi.org/10.14419/ijet.v7i4.10.26660.
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In this paper we develop on abstract system: viz Boolean-like algebra and prove that every Boolean algebra is a Boolean-like algebra. A necessary and sufficient condition for a Boolean-like algebra to be a Boolean algebra has been obtained. As in the case of Boolean ring and Boolean algebra, it is established that under suitable binary operations the Boolean-like ring and Boolean-like algebra are equivalent abstract structures.
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Nerode, A., and J. B. Remmel. "Complexity-theoretic algebra II: Boolean algebras." Annals of Pure and Applied Logic 44, no. 1-2 (October 1989): 71–99. http://dx.doi.org/10.1016/0168-0072(89)90047-x.
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S, Vijayabarathi, and Srinivasa Rao K. "Boolean Algebras of Pre A*-Algebra." International Journal of Mathematics Trends and Technology 7, no. 1 (March 25, 2014): 61–67. http://dx.doi.org/10.14445/22315373/ijmtt-v7p509.
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Szeto, George, and Lianyong Xue. "The Boolean algebra of Galois algebras." International Journal of Mathematics and Mathematical Sciences 2003, no. 11 (2003): 673–79. http://dx.doi.org/10.1155/s0161171203202210.
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LetBbe a Galois algebra with Galois groupG,Jg={b∈B|bx=g(x)b for all x∈B}for eachg∈G, andBJg=Begfor a central idempotenteg,Bathe Boolean algebra generated by{0,eg|g∈G},ea nonzero element inBa, andHe={g∈G|eeg=e}. Then, a monomialeis characterized, and the Galois extensionBe, generated byewith Galois groupHe, is investigated.
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MOVSISYAN, YU M., and V. A. ASLANYAN. "SUPER-BOOLEAN FUNCTIONS AND FREE BOOLEAN QUASILATTICES." Discrete Mathematics, Algorithms and Applications 06, no. 02 (March 19, 2014): 1450024. http://dx.doi.org/10.1142/s1793830914500244.
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A Boolean quasilattice is an algebra with hyperidentities of the variety of Boolean algebras. In this paper, we give a functional representation of the free n-generated Boolean quasilattice with two binary, one unary and two nullary operations. Namely, we define the concept of super-Boolean function and prove that the free Boolean quasilattice with two binary, one unary and two nullary operations on n free generators is isomorphic to the Boolean quasilattice of super-Boolean functions of n variables.
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LEMPP, STEFFEN, MIKHAIL PERETYAT'KIN, and REED SOLOMON. "THE LINDENBAUM ALGEBRA OF THE THEORY OF THE CLASS OF ALL FINITE MODELS." Journal of Mathematical Logic 02, no. 02 (November 2002): 145–225. http://dx.doi.org/10.1142/s0219061302000163.
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In this paper, we investigate the Lindenbaum algebra ℒ(T fin ) of the theory T fin = Th (M fin ) of the class M fin of all finite models of a finite rich signature. We prove that this algebra is an atomic Boolean algebra while its Gödel numeration γ is a [Formula: see text]-numeration. Moreover, the quotient algebra (ℒ(T fin )/ℱ, γ/ℱ) modulo the Fréchet ideal ℱ is a [Formula: see text]-algebra, which is universal over the class of all [Formula: see text] Boolean algebras. These conditions characterize uniquely the algebra ℒ(T fin ); moreover, these conditions characterize up to recursive isomorphism the numerated Boolean quotient algebra (ℒ(T fin )/ℱ, γ/ℱ). These results extend the work of Trakhtenbrot [17] and Vaught [18] on the first order theory of the class of all finite models of a finite rich signature.
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Kudryavtseva, Ganna, and Jonathan Leech. "Free skew Boolean algebras." International Journal of Algebra and Computation 26, no. 07 (November 2016): 1323–48. http://dx.doi.org/10.1142/s0218196716500569.
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We study the structure and properties of free skew Boolean algebras (SBAs). For finite generating sets, these free algebras are finite and we give their representation as a product of primitive algebras and provide formulas for calculating their cardinality. We also characterize atomic elements and central elements, and calculate the number of such elements. These results are used to study minimal generating sets of finite SBAs. We also prove that the center of the free infinitely generated algebra is trivial and show that all free algebras have intersections.
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Banaschewski, B., and K. R. Bhutani. "Boolean algebras in a localic topos." Mathematical Proceedings of the Cambridge Philosophical Society 100, no. 1 (July 1986): 43–55. http://dx.doi.org/10.1017/s0305004100065853.
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When a familiar notion is modelled in a certain topos E, the natural problem arises to what extent theorems concerning its models in usual set theory remain valid for its models in E, or how suitable properties of E affect the validity of certain of these theorems. Problems of this type have in particular been studied by Banaschewski[2], Bhutani[5], and Ebrahimi[6, 7], dealing with abelian groups in a localic topos and universal algebra in an arbitrary Grothendieck topos. This paper is concerned with Boolean algebras, specifically with injectivity and related topics for the category of Boolean algebras in the topos of sheaves on a locale and with properties of the initial Boolean algebra in .
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Celani, Sergio. "Generalized join-hemimorphisms on Boolean algebras." International Journal of Mathematics and Mathematical Sciences 2003, no. 11 (2003): 681–93. http://dx.doi.org/10.1155/s016117120311112x.
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We introduce the notions of generalized join-hemimorphism and generalized Boolean relation as an extension of the notions of join-hemimorphism and Boolean relation, respectively. We prove a duality between these two notions. We will also define a generalization of the notion of Boolean algebra with operators by considering a finite family of Boolean algebras endowed with a generalized join-hemimorphism. Finally, we define suitable notions of subalgebra, congruences, Boolean equivalence, and open filters.
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Adams, M. E. "Principal congruences in de Morgan algebras." Proceedings of the Edinburgh Mathematical Society 30, no. 3 (October 1987): 415–21. http://dx.doi.org/10.1017/s0013091500026808.
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A congruence relation θ on an algebra L is principal if there exist a, b)∈L such that θ is the smallest congruence relation for which (a, b)∈θ. The property that, for every algebra in a variety, the intersection of two principal congruences is again a principal congruence is one that is known to be shared by many varieties (see, for example, K. A. Baker [1]). One such example is the variety of Boolean algebras. De Morgan algebras are a generalization of Boolean algebras and it is the intersection of principal congruences in the variety of de Morgan algebras that is to be considered in this note.
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FLAUT, CRISTINA. "Some applications of idempotent elements in MV algebras." Carpathian Journal of Mathematics 39, no. 1 (July 30, 2022): 161–74. http://dx.doi.org/10.37193/cjm.2023.01.10.
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"In this paper we provide some properties and applications of MV-algebras. We prove that a Fibonacci stationary sequence in an MV-algebra gives us an idempotent element. Moreover, taking into account of the representation of a finite MV-algebra, by using Boolean elements of this algebra, we prove that a Fibonacci sequence in an MV-algebra is always stationary. This result is interesting comparing with the behavior of such a sequence on the group $\left( \mathbb{Z}% _{n},+\right) $, where the Fibonacci sequences are periodic, with the period given by the Pisano period. We also give some examples of finite MV-algebras and the number of their idempotent elements. As an application in Coding Theory, to a Boolean algebra it is attached a binary block code and it is proved that, under some conditions, the converse is also true."
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Dorninger, Dietmar, and Helmut Länger. "On ring-like structures of lattice-ordered numerical events." Asian-European Journal of Mathematics 14, no. 10 (March 12, 2021): 2150186. http://dx.doi.org/10.1142/s1793557121501862.
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Let [Formula: see text] be a set of states of a physical system. The probabilities [Formula: see text] of the occurrence of an event when the system is in different states [Formula: see text] define a function from [Formula: see text] to [Formula: see text] called a numerical event or, more accurately, an [Formula: see text]-probability. Sets of [Formula: see text]-probabilities ordered by the partial order of functions give rise to so-called algebras of [Formula: see text]-probabilities, in particular, to the ones that are lattice-ordered. Among these, there are the [Formula: see text]-algebras known from probability theory and the Hilbert-space logics which are important in quantum-mechanics. Any algebra of [Formula: see text]-probabilities can serve as a quantum-logic, and it is of special interest when this logic turns out to be a Boolean algebra because then the observed physical system will be classical. Boolean algebras are in one-to-one correspondence to Boolean rings, and the question arises to find an analogue correspondence for lattice-ordered algebras of [Formula: see text]-probabilities generalizing the correspondence between Boolean algebras and Boolean rings. We answer this question by defining ring-like structures of events (RLSEs). First, the structure of RLSEs is revealed and Boolean rings among RLSEs are characterized. Then we establish how RLSEs correspond to lattice-ordered algebras of numerical events. Further, functions for associating lattice-ordered algebras of [Formula: see text]-probabilities to RLSEs are studied. It is shown that there are only two ways to assign lattice-ordered algebras of [Formula: see text]-probabilities to RLSEs if one restricts the corresponding mappings to term functions over the underlying orthomodular lattice. These term functions are the very functions by which also the Boolean algebras can be assigned to Boolean rings.
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PINTER, CHARLES C. "STONE SPACE OF CYLINDRIC ALGEBRAS AND TOPOLOGICAL MODEL SPACES." Journal of Symbolic Logic 81, no. 3 (September 2016): 1069–86. http://dx.doi.org/10.1017/jsl.2016.11.
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AbstractThe Stone representation theorem was a milestone for the understanding of Boolean algebras. From Stone’s theorem, every Boolean algebra is representable as a field of sets with a topological structure. By means of this, the structural elements of any Boolean algebra, as well as the relations between them, are represented geometrically and can be clearly visualized. It is no different for cylindric algebras: Suppose that ${\frak A}$ is a cylindric algebra and ${\cal S}$ is the Stone space of its Boolean part. (Among the elements of the Boolean part are the diagonal elements.) It is known that with nothing more than a family of equivalence relations on ${\cal S}$ to represent quantifiers, ${\cal S}$ represents the full cylindric structure just as the Stone space alone represents the Boolean structure. ${\cal S}$ with this structure is called a cylindric space.Many assertions about cylindric algebras can be stated in terms of elementary topological properties of ${\cal S}$. Moreover, points of ${\cal S}$ may be construed as models, and on that construal ${\cal S}$ is called a model space. Certain relations between points on this space turn out to be morphisms between models, and the space of models with these relations hints at the possibility of an “abstract” model theory. With these ideas, a point-set version of model theory is proposed, in the spirit of pointless topology or category theory, in which the central insight is to treat the semantic objects (models) homologously with the corresponding syntactic objects so they reside together in the same space.It is shown that there is a new, purely algebraic way of introducing constants in cylindric algebras, leading to a simplified proof of the representation theorem for locally finite cylindric algebras. Simple rich algebras emerge as homomorphic images of cylindric algebras. The topological version of this theorem is especially interesting: The Stone space of every locally finite cylindric algebra ${\frak A}$ can be partitioned into subspaces which are the Stone spaces of all the simple rich homomorphic images of ${\frak A}$. Each of these images completely determines a model of ${\frak A}$, and all denumerable models of ${\frak A}$ appear in this representation.The Stone space ${\cal S}$ of every cylindric algebra can likewise be partitioned into closed sets which are duals of all the types in ${\frak A}$. This fact yields new insights into miscellaneous results in the model theory of saturated models.
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Szeto, George, and Lianyong Xue. "The Boolean algebra and central Galois algebras." International Journal of Mathematics and Mathematical Sciences 28, no. 4 (2001): 237–42. http://dx.doi.org/10.1155/s0161171201007104.
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LetBbe a Galois algebra with Galois groupG,Jg={b∈B∣bx=g(x)b for all x∈B}forg∈G, andBJg=Begfor a central idempotenteg. Then a relation is given between the set of elements in the Boolean algebra(Ba,≤)generated by{0,eg∣g∈G}and a set of subgroups ofG, and a central Galois algebraBewith a Galois subgroup ofGis characterized for ane∈Ba.
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Harris, Kenneth, and Antonio Montalbán. "Boolean algebra approximations." Transactions of the American Mathematical Society 366, no. 10 (June 3, 2014): 5223–56. http://dx.doi.org/10.1090/s0002-9947-2014-05950-3.
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Chernoskutov, Yu Yu. "On the Syllogistic of G. Boole." Discourse 7, no. 2 (April 29, 2021): 5–15. http://dx.doi.org/10.32603/2412-8562-2021-7-2-5-15.
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Introduction. This article focuses on the investigation of Boole’s theory of categorical syllogism, exposed in his book “The Mathematical analysis of Logic”. That part of Boolean legacy has been neglected in the prevailed investigations on the history of logic; the latter provides the novelty of the work presented.Methodology and sources. The formal reconstruction of the methods of algebraic presentation of categorical syllogism, as it is exposed in the original work of Boole, is conducted. The character of Boolean methods is investigated in the interconnections with the principles of symbolic algebra on the one hand, and with the principles of signification, taken from R. Whately, on the other hand. The approaches to signification, grounding the syllogistic theories of Boole and Brentano, are analyzed in comparison, wherefrom we explain the reasons why the results of those theories are different so much.Results and discussion. It is demonstrated here that Boole has borrowed the principles of signification from the Whately’s book “The Elements of Logic”. The interpreting the content of the terms as classes, being combined with methods of symbolic algebra, has determined the core features of Boolean syllogism theory and its unexpected results. In contrast to Whately, Boole conduct the approach to ultimate ends, overcoming the restrictions imposed by Aristotelean doctrine. In particular, he neglects the distinction of subject and predicate among the terms of proposition, the order of premises, and provide the possibility to draw conclusions with negative terms. At the same time Boole missed that the forms of inference, parallel to Bramantip and Fresison, are legitimate forms in his system. In spite of the apparent affinities between the Boolean and Brentanian theories of judgment, the syllogistics of Boole appeared to be more flexible. The drawing of particular conclusion from universal premises is allowable in Boolean theory, but not in Brentanian one; besides, in his theory is allowable the drawing of conclusion from two negative premises, which is prohibited in Aristotelian syllogistic.Conclusion. Boole consistently interpreted signification of terms as classes; being combine with methods symbolic algebra it led to very flexible syllogism theory with rich results.
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Meng, Biao Long, and Xiao Long Xin. "A Note of Filters in Effect Algebras." Chinese Journal of Mathematics 2013 (November 10, 2013): 1–4. http://dx.doi.org/10.1155/2013/570496.
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We investigate relations of the two classes of filters in effect algebras (resp., MV-algebras). We prove that a lattice filter in a lattice ordered effect algebra (resp., MV-algebra) does not need to be an effect algebra filter (resp., MV-filter). In general, in MV-algebras, every MV-filter is also a lattice filter. Every lattice filter in a lattice ordered effect algebra is an effect algebra filter if and only if is an orthomodular lattice. Every lattice filter in an MV-algebra is an MV-filter if and only if is a Boolean algebra.
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Terwijn, Sebastiaan A. "On the structure of the Medvedev lattice." Journal of Symbolic Logic 73, no. 2 (June 2008): 543–58. http://dx.doi.org/10.2178/jsl/1208359059.
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AbstractWe investigate the structure of the Medvedev lattice as a partial order. We prove that every interval in the lattice is either finite, in which case it is isomorphic to a finite Boolean algebra, or contains an antichain of size . the size of the lattice itself. We also prove that it is consistent with ZFC that the lattice has chains of size . and in fact that these big chains occur in every infinite interval. We also study embeddings of lattices and algebras. We show that large Boolean algebras can be embedded into the Medvedev lattice as upper semilattices, but that a Boolean algebra can be embedded as a lattice only if it is countable. Finally we discuss which of these results hold for the closely related Muchnik lattice.
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Hehner, Eric C. R. "From boolean algebra to unified algebra." Mathematical Intelligencer 26, no. 2 (March 2004): 3–19. http://dx.doi.org/10.1007/bf02985647.
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Bell, J. L. "Zorn's lemma and complete Boolean algebras in intuitionistic type theories." Journal of Symbolic Logic 62, no. 4 (December 1997): 1265–79. http://dx.doi.org/10.2307/2275642.
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AbstractWe analyze Zorn's Lemma and some of its consequences for Boolean algebras in a constructive setting. We show that Zorn's Lemma is persistent in the sense that, if it holds in the underlying set theory, in a properly stated form it continues to hold in all intuitionistic type theories of a certain natural kind. (Observe that the axiom of choice cannot be persistent in this sense since it implies the law of excluded middle.) We also establish the persistence of some familiar results in the theory of (complete) Boolean algebras—notably, the proposition that every complete Boolean algebra is an absolute subretract. This (almost) resolves a question of Banaschewski and Bhutani as to whether the Sikorski extension theorem for Boolean algebras is persistent.
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Mckenzie, Ralph, and J. Donald Monk. "On some small cardinals for Boolean algebras." Journal of Symbolic Logic 69, no. 3 (September 2004): 674–82. http://dx.doi.org/10.2178/jsl/1096901761.
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Abstract.Assume that all algebras are atomless. (1) Spind(A × B) = Spind(A) ∪ Spind(B). (2) Spind(Ai). Now suppose that κ and λ are infinite cardinals, with κ uncountable and regular and with κ < λ. (3) There is an atomless Boolean algebra A such that u(A) = κ and i(A) = λ. (4) If λ is also regular, then there is an atomless Boolean algebra A such that t(A) = s(A) = κ and α (A) = λ. All results are in ZFC, and answer some problems posed in Monk [01] and Monk [∞].
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Chajda, Ivan, Helmut Länger, and Ranganathan Padmanabhan. "Single identities forcing lattices to be Boolean." Mathematica Slovaca 68, no. 4 (August 28, 2018): 713–16. http://dx.doi.org/10.1515/ms-2017-0138.
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Abstract In this note we characterize Boolean algebras among lattices of type (2, 2, 1) with join, meet and an additional unary operation by means of single two-variable respectively three-variable identities. In particular, any uniquely complemented lattice satisfying any one of these equational constraints is distributive and hence a Boolean algebra.
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Jockusch, Carl G., and Robert I. Soare. "Boolean algebras, Stone spaces, and the iterated Turing jump." Journal of Symbolic Logic 59, no. 4 (December 1994): 1121–38. http://dx.doi.org/10.2307/2275695.
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AbstractWe show, roughly speaking, that it requires ω iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if α is a recursive ordinal, is a countable structure with finite signature, and d is a degree, we say that has αth-jump degreed if d is the least degree which is the αth jump of some degree c such there is an isomorphic copy of with universe ω in which the functions and relations have degree at most c. We show that every degree d ≥ 0(ω) is the ωth jump degree of a Boolean algebra, but that for n < ω no Boolean algebra has nth-jump degree d < 0(n). The former result follows easily from work of L. Feiner. The proof of the latter result uses the forcing methods of J. Knight together with an analysis of various equivalences between Boolean algebras based on a study of their Stone spaces. A byproduct of the proof is a method for constructing Stone spaces with various prescribed properties.
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Hansen, Lars. "On an algebra of lattice-valued logic." Journal of Symbolic Logic 70, no. 1 (March 2005): 282–318. http://dx.doi.org/10.2178/jsl/1107298521.
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AbstractThe purpose of this paper is to present an algebraic generalization of the traditional two-valued logic. This involves introducing a theory of automorphism algebras, which is an algebraic theory of many-valued logic having a complete lattice as the set of truth values. Two generalizations of the two-valued case will be considered, viz., the finite chain and the Boolean lattice. In the case of the Boolean lattice, on choosing a designated lattice value, this algebra has binary retracts that have the usual axiomatic theory of the propositional calculus as suitable theory. This suitability applies to the Boolean algebra of formalized token models [2] where the truth values are, for example, vocabularies. Finally, as the actual motivation for this paper, we indicate how the theory of formalized token models [2] is an example of a many-valued predicate calculus.
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Kumar, V. Venkata, and M. Sambasiva Rao. "Dual annihilator filters of commutative BE-algebras." Asian-European Journal of Mathematics 10, no. 01 (March 2017): 1750013. http://dx.doi.org/10.1142/s1793557117500139.
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Some properties of dual annihilator filters of commutative [Formula: see text]-algebras are studied. It is proved that the class of all dual annihilator filters of a BE-algebra is a complete Boolean algebra. A set of equivalent conditions is derived for every prime filter of a commutative [Formula: see text]-algebra to become a maximal filter.
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Estaji, Ali, and Toktam Haghdadi. "The pointfree version of grills." Filomat 34, no. 8 (2020): 2667–81. http://dx.doi.org/10.2298/fil2008667e.
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In this paper, the pointfree version of grills is introduced. We consider a Boolean algebra B and a subframe L instead of a topological space (X,?), and present the concept of approximation ? over B. Moreover, some properties of them are given. Also, we introduce and study the new concepts grill and ?-grill on Boolean algebras.
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Aizpuru, A., and A. Gutiérrez-Dávila. "On Orlicz-Pettis property." Studia Scientiarum Mathematicarum Hungarica 40, no. 1-2 (July 1, 2003): 171–81. http://dx.doi.org/10.1556/sscmath.40.2003.1-2.14.
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In this paper we study the Orlicz-Pettis property for a Boolean algebra. We characterize the countable Boolean algebras with this property and extend that study to some families of P(N). As a consequence, we obtain characterizations of weakly summing families in terms of the space c0 and in terms of some separation properties.
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NURAKUNOV, ANVAR M., and MICHAŁ M. STRONKOWSKI. "PROFINITENESS IN FINITELY GENERATED VARIETIES IS UNDECIDABLE." Journal of Symbolic Logic 83, no. 04 (December 2018): 1566–78. http://dx.doi.org/10.1017/jsl.2017.89.
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AbstractProfinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety ${\cal V}$ is standard if every Boolean topological algebra with the algebraic reduct in ${\cal V}$ is profinite.We show that there is no algorithm which takes as input a finite algebra A of a finite type and decide whether the variety $V\left( {\bf{A}} \right)$ generated by A is standard. We also show the undecidability of some related properties. In particular, we solve a problem posed by Clark, Davey, Freese, and Jackson.We accomplish this by combining two results. The first one is Moore’s theorem saying that there is no algorithm which takes as input a finite algebra A of a finite type and decides whether $V\left( {\bf{A}} \right)$ has definable principal subcongruences. The second is our result saying that possessing definable principal subcongruences yields possessing finitely determined syntactic congruences for varieties. The latter property is known to yield standardness.
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Ozawa, Masanao. "Boolean valued interpretation of Banach space theory and module structures of von Neumann algebras." Nagoya Mathematical Journal 117 (March 1990): 1–36. http://dx.doi.org/10.1017/s0027763000001793.
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Recently, systematic applications of the Scott-Solovay Boolean valued set theory were done by several authors; Takeuti [25, 26, 27, 28, 29, 30], Nishimura [13, 14] Jech [8] and Ozawa [15, 16, 17, 18, 19, 20] in analysis and Smith [23], Eda [2, 3] in algebra. This approach seems to be providing us with a new and powerful machinery in analysis and algebra. In the present paper, we shall study Banach space objects in the Scott-Solovay Boolean valued universe and provide some useful transfer principles from theorems of Banach spaces to theorems of Banach modules over commutative AW*-algebras. The obtained machinery will be applied to resolve some problems concerning the module structures of von Neumann algebras.
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Downey, Rod, and Carl G. Jockusch. "Effective presentability of Boolean algebras of Cantor-Bendixson rank 1." Journal of Symbolic Logic 64, no. 1 (March 1999): 45–52. http://dx.doi.org/10.2307/2586749.
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AbstractWe show that there is a computable Boolean algebra and a computably enumerable ideal I of such that the quotient algebra /I is of Cantor-Bendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though Feiner's construction yields a Boolean algebra of infinite Cantor-Bendixson rank.
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Westerbaan, Abraham, Bas Westerbaan, and John van de Wetering. "The three types of normal sequential effect algebras." Quantum 4 (December 24, 2020): 378. http://dx.doi.org/10.22331/q-2020-12-24-378.
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A sequential effect algebra (SEA) is an effect algebra equipped with a sequential product operation modeled after the Lüders product (a,b)↦aba on C∗-algebras. A SEA is called normal when it has all suprema of directed sets, and the sequential product interacts suitably with these suprema. The effects on a Hilbert space and the unit interval of a von Neumann or JBW algebra are examples of normal SEAs that are in addition convex, i.e. possess a suitable action of the real unit interval on the algebra. Complete Boolean algebras form normal SEAs too, which are convex only when 0=1.We show that any normal SEA E splits as a direct sum E=Eb⊕Ec⊕Eac of a complete Boolean algebra Eb, a convex normal SEA Ec, and a newly identified type of normal SEA Eac we dub purely almost-convex.Along the way we show, among other things, that a SEA which contains only idempotents must be a Boolean algebra; and we establish a spectral theorem using which we settle for the class of normal SEAs a problem of Gudder regarding the uniqueness of square roots. After establishing our main result, we propose a simple extra axiom for normal SEAs that excludes the seemingly pathological a-convex SEAs. We conclude the paper by a study of SEAs with an associative sequential product. We find that associativity forces normal SEAs satisfying our new axiom to be commutative, shedding light on the question of why the sequential product in quantum theory should be non-associative.