Primes in the semigroup of Boolean matrices

Caen, D. de; Gregory, David A.

doi:10.1016/0024-3795(81)90172-5

Cited by 28 publications

(7 citation statements)

References 8 publications

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“…De Caen and Gregory [13] showed that if α ∈ B n is prime then no column of α can contain another, and no row of α can contain another row. In particular if α ∈ B n is prime then α has no zero row or column and no row or column with all entries equal to 1.…”

Section: 3mentioning

confidence: 99%

The lattice and semigroup structure of multipermutations

Carvalho¹,

Martin²

2021

Preprint

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We study the algebraic properties of binary relations whose underlying digraph is smooth, that is has no source or sink. Such objects have been studied as surjective hyper-operations (shops) on the corresponding vertex set, and as binary relations that are defined everywhere and whose inverse is also defined everywhere. In the latter formulation, they have been called multipermutations.We study the lattice structure of sets (monoids) of multipermutations over an n-element domain. Through a Galois connection, these monoids form the algebraic counterparts to sets of relations closed under definability in positive first-order logic without equality. The first side of this Galois connection has been elaborated previously, we show the other side here. We study the property of inverse on multipermutations and how it connects our monoids to groups. We use our results to give a simple dichotomy theorem for the evaluation problem of positive first-order logic without equality on the class of structures whose preserving multipermutations form a monoid closed under inverse. These problems turn out either to be in Logspace or to be Pspacecomplete. We go on to study the monoid of all multipermutations on an n-element domain, under usual composition of relations. We characterise its Green relations, regular elements and show that it does not admit a generating set that is polynomial on n.

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Section: 3mentioning

confidence: 99%

The lattice and semigroup structure of multipermutations

Carvalho¹,

Martin²

2021

Preprint

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“…If a D-class contains a regular element then every element of that class is regular. An element x ∈ B 3 is called prime if it is not a permutation matrix and whenever x = yz either x or y is a permutation matrix; see de Caen and Gregory [4]. If a D-class contains a prime element then every element of that class is prime.…”

Section: D-classesmentioning

confidence: 99%

“…We calculate the dimension of the two-sided ideal of S generated by each element; since these elements are central, it suffices to consider all left multiples by basis elements of S. We obtain the values 1,1,1,4,9,9,9,9,36,36,49,81,81,144 for the elements in the order of Table 7. We have reordered the basis elements of the center so that these dimensions are non-decreasing.…”

Section: The Decomposition Into Simple Idealsmentioning

confidence: 99%

Structure of the rational monoid algebra for Boolean matrices of order 3

Bremner

2014

Linear Algebra and its Applications

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We use computer algebra to study the 512-dimensional associative algebra QB 3 , the rational monoid algebra of 3×3 Boolean matrices. We obtain a basis for the radical in bijection with the 42 non-regular elements of B 3 . The center of the 470-dimensional semisimple quotient has dimension 14; we use a splitting algorithm to find a basis of orthogonal primitive idempotents. We show that the semisimple quotient is the direct sum of simple two-sided ideals isomorphic to matrix algebras M d (Q) for d = 1, 1, 1, 2, 3, 3, 3, 3, 6, 6, 7, 9, 9, 12. We construct the irreducible representations of B 3 over Q by calculating the representation matrices for a minimal set of generators.

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“…the system of linear equations Ax = b is solvable, then rank d (A) = rank d (A, b), where (A, b) = A(1) , A(2) , . .…”

mentioning

confidence: 99%