Degree of matrix relation algebras

Bachraoui, Mohamed El; Vel, M.L.J. van de

doi:10.1007/s000120200002

Cited by 3 publications

(7 citation statements)

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“…The following hold in any MVRA (in the case of points (2) and (6) we assume that the suprema in the left-hand sides of the equalities exist): Vol. 56, 2007 Some algebraic theory for many-valued relation algebras 215…”

Section: Definition and Basic Properties Of Mv-relation-algebrasmentioning

confidence: 99%

“…. , n} and find the matrix relation algebras studied in the Boolean case in [1] and [2] (see Section 8).…”

Section: Examplesmentioning

confidence: 99%

“…(1) ≤ is a lattice order on A, with sup = ∨ and inf = ∧, (2) x ≤ y iff x ⊕ y = 1, (3) x ⊕ x = 1, (4) x x = 0, (5) x ≤ y iff y ≤ x, (6) x y ≤ z iff x ≤ y ⊕ z, 214 A. Popescu Algebra univers. (7) z ≤ x ⊕ y iff z x ≤ y, (8) x i∈I y i = i∈I (x y i ), (9) x ⊕ i∈I y i = i∈I (x ⊕ y i ), (10) d(x, y) = 0 iff x = y, (11) d(x, y) = d(y, x), (12) d(x, z) ≤ d(x, y) ⊕ d(y, z), (13) d(x, y) = d(x, y), (14) …”

Section: Introductionmentioning

confidence: 99%

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Some algebraic theory for many-valued relation algebras

Popescu

2007

Algebra univers.

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We study MV-relation-algebras, appearing by abstracting away from the concrete many-valued relations and the operations on them, such as composition and converse. MV-relation-algebras are MV generalizations of the relation algebras developed by A. Tarski and his school starting from the late forties. Some facts about ideals, congruences, and various types of elements are proved. A characterization of the "natural" MV-relation-algebras (a parameterized analogue of the classical full proper relation algebras) is also provided, as well as a first-order elementary description of matrix MV-relation algebras.

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Section: Definition and Basic Properties Of Mv-relation-algebrasmentioning

confidence: 99%

“…. , n} and find the matrix relation algebras studied in the Boolean case in [1] and [2] (see Section 8).…”

Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some algebraic theory for many-valued relation algebras

Popescu

2007

Algebra univers.

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“…As a consequence of Lemma 4.11 (1,3), observe that K ii is an equivalence element for all i < n, and even a simple one by Lemma 4.11(5). Now assume that i, k < n with i = k. Clearly K ij ≤ I ii ; 1 ≤ J i ; 1 and K kl ≤ I kk ; 1 ≤ J k ; 1.…”

Section: El Bachraoui and M Van De Velmentioning

confidence: 99%

“…3.5 for all n > 0. In addition, M n (R) is non-integral by Remark 3.6 and if R is non-representable then so is M n (R) by [3,Thm. 2.5].…”

Section: Theoremmentioning

confidence: 99%

Matrix relation algebras

Bachraoui¹,

Vel²

2002

Algebra Universalis

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Square matrices over a relation algebra are relation algebras in a natural way. We show that for fixed n, these algebras can be characterized as reducts of some richer kind of algebra. Hence for fixed n, the class of n × n matrix relation algebras has a first-order characterization. As a consequence, homomorphic images and proper extensions of matrix relation algebras are isomorphic to matrix relation algebras.

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Łukasiewicz-Moisil Relation Algebras

Popescu

2005

Stud Logica

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We introduce Lukasiewicz-Moisil relation algebras, obtained by considering a relational dimension over Lukasiewicz-Moisil algebras. We prove some arithmetical properties, provide a characterization in terms of complex algebras, study the connection with relational Post algebras and characterize the simple structures and the matrix relation algebras. A. Popescu[11] and to the MV approach from [32] and [33]. Unlike in these latter cases, the connection with classical logic, being explicitly stated by means of the Chrysippian nuances, allows one to point out certain interesting similarities between LM and classical relational structures. Therefore, we consider that an abstract study of LM-based relation algebras is useful, not only for enriching the many-valued relational algebraic theory developed in [11,32,33], but also for raising the issue of communication with classical relational structures. In this paper, we intend to study these structures, using the following route.Section 2 is a preliminary section on Lukasiewicz-Moisil algebras and classical relation algebras. In Section 3, we introduce LM-relation algebras (LMRAs) and prove some arithmetical properties of these structures. Section 4 studies LM-based complex algebras. There are proved a characterization of complex algebras, as well as a representation theorem similar to the one from the classical case. In Section 5, using the adjunction to Boolean relation algebras preserved from the "non-relational" case, we prove some structural properties of LMRAs -among them, the fact that "simple" is equivalent to "subdirectly irreducible" and a representation theorem for concrete LMRAs, that is, LMRAs isomorphic to concrete structures of many-valued relations on some set X. In Section 6, we study matrix relation algebras and prove that they form an elementary class. A brief discussion of related work ends the paper.

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Degree of matrix relation algebras

Abstract: We show that a relation algebra and its n-matrix relation algebra have the same degree for all positive integers n. An intermediate result relates the degree of a relation algebra to the degree of a relativization with respect to equivalence elements.

Cited by 3 publications

References 4 publications

Some algebraic theory for many-valued relation algebras

Some algebraic theory for many-valued relation algebras

Matrix relation algebras

Łukasiewicz-Moisil Relation Algebras

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