Boolean inner-product spaces and Boolean matrices

Gudder, Stan; Latrémolière, Frédéric

doi:10.1016/j.laa.2009.02.028

Cited by 8 publications

(2 citation statements)

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“…The map x → x (rk) is bijective because (x (rk) ) (rk) = x by Lemma 4.8 (6) and (B4). The map is a homomorphism: t r (x, y, z) (rk) = L.4.8 (7) t r (x, y (rk) , z (rk) ) = t r ((x (rk) ) (rk) , y (rk) , z (rk) ) = q((x (rk) ) (rk) , y (rk) /r, z (rk) /r) = L.4.8 (4) q(x (rk) , y (rk) /k, z (rk) /k) = t k (x (rk) , y (rk) , z (rk) ). Moreover, (e r ) (rk) = e (rk)r = e k .…”

Section: 2mentioning

confidence: 99%

On noncommutative generalisations of Boolean algebras

Bucciarelli

Salibra

2019

Art Discrete Appl. Math.

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Skew Boolean algebras (skew BA) and Boolean-like algebras (nBA) are onepointed and n-pointed noncommutative generalisation of Boolean algebras, respectively. We show that any nBA is a cluster of n isomorphic right-handed skew BAs, axiomatised here as the variety of skew star algebras. The variety of skew star algebras is shown to be term equivalent to the variety of nBAs. We use skew BAs in order to develop a general theory of multideals for nBAs. We also provide a representation theorem for right-handed skew BAs in terms of nBAs of n-partitions.Date: May 30, 2019.

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Section: 2mentioning

confidence: 99%

On noncommutative generalisations of Boolean algebras

Bucciarelli

Salibra

2019

Art Discrete Appl. Math.

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“…An R-semimodule is called: (i) a module if R is a ring; (ii) a vector space if R is a field; (iii) a Boolean vector space (see [9] for basic facts on Boolean vector spaces) if R is a Boolean algebra. The elements of an R-semimodule are called vectors.…”

Section: 2mentioning

confidence: 99%

Boolean-like algebras of finite dimension

Bucciarelli¹,

Ledda²,

Paoli³

et al. 2018

Preprint

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We introduce Boolean-like algebras of dimension n (nBA) having n constants e 1 , . . . , en, and an (n + 1)-ary operation q (a "generalised if-then-else") that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of nBAs share many remarkable properties with the variety of Boolean algebras and with primal varieties. Exploiting the concept of central element, we extend the notion of Boolean power to that of semiring power and we prove two representation theorems: (i) Any pure nBA is isomorphic to the algebra of n-central elements of a Boolean vector space; (ii) Any member of a variety of nBAs with one generator is isomorphic to a Boolean power of this generator. This gives a new proof of Foster's theorem on primal varieties. The nBAs provide the algebraic framework for generalising the classical propositional calculus to the case of n -perfectly symmetric -truth-values. Every finite-valued tabular logic can be embedded into such an n-valued propositional logic, nCL, and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in nCL, and, via the embeddings, in all the finite tabular logics.

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