Invertible matrices over distributive lattices

In this paper we characterize those linear bijective maps on the monoid of all n × n square matrices over an anti-negative semifield which preserve and strongly preserve each of Green's equivalence relations L, R, D, J and the corresponding three pre-orderings ≤L, ≤R, ≤J . These results apply in particular to the tropical and boolean semirings, and for these two semirings we also obtain corresponding results for the H relation.

Section: Bijective Linear Transformations Preserving Green's Relationsmentioning

confidence: 99%

Linear isomorphisms preserving Green's relations for matrices over anti-negative semifields

Guterman

Johnson

Kambites

2018

“…Corollary 3.3 [13]. If D is a distributive lattice and A = (a ij ) ∈ M n (D), then the following statements are equivalent to each other:…”

Section: Proof It Follows Immediately From Theorem 34mentioning

confidence: 76%

“…Theorem 3.4 generalizes Theorem 1 of Skornyakov[13]. If A ∈ M n (L) is an invertible matrix, then f A is an injective mapping.…”

mentioning

confidence: 71%

Invertible incline matrices and Cramer's rule over inclines

Han

2004

Inclines are the additively idempotent semirings in which the products are less than or equal to factors. Thus inclines generalize Boolean algebra, fuzzy algebra and distributive lattice. And the Boolean matrices, the fuzzy matrices and the lattice matrices are the prototypical examples of the incline matrices (i.e., the matrices over inclines). In this paper, the complete description of the invertible incline matrices is given. Some necessary and sufficient conditions for an incline matrix to be invertible are studied, Cramer's rule over inclines is presented and the group of invertible incline matrices is investigated. The main results in the present paper generalize and develop the corresponding results in the literatures for the Boolean matrices, the fuzzy matrices and the lattice matrices.

“…Invertibility of Boolean matrices was studied in [8,9,13,21] and the following result is well-known. We present here a short proof which relies upon our previous work with orthonormal bases and generalize the invertibility result to show that invertible rectangular matrices have to be square.…”

Section: Stochastic and Unitary Matricesmentioning

confidence: 99%

Boolean inner-product spaces and Boolean matrices

Gudder

Latrémolière

2009

This article discusses the concept of Boolean spaces endowed with a Boolean valued inner product and their matrices. A natural inner product structure for the space of Boolean n-tuples is introduced. Stochastic boolean vectors and stochastic and unitary Boolean matrices are studied. A dimension theorem for orthonormal bases of a Boolean space is proven. We characterize the invariant stochastic Boolean vectors for a Boolean stochastic matrix and show that they can be used to reduce a unitary matrix. Finally, we obtain a result on powers of stochastic and unitary matrices.