Division semirings with 1+1=1

Hutchins, Harry

doi:10.1007/bf02572796

Cited by 10 publications

(2 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(In fact, every idempotent semifield arises in this way -see Lemma 2.1 below.) For ease of reference we record (with no claim of originality) a number of facts about semifields, which have been observed by many authors (see for example [15,19,44]): Lemma 2.1. Let S be a semifield.…”

Section: 1mentioning

confidence: 99%

Matrix semigroups over semirings

Gould

Johnson

Naz

2019

Int. J. Algebra Comput.

View full text Add to dashboard Cite

The multiplicative semigroup Mn(F ) of n × n matrices over a field F is well understood, in particular, it is a regular semigroup. This paper considers semigroups of the form Mn(S), where S is a semiring, and the subsemigroups U Tn(S) and Un(S) of Mn(S) consisting of upper triangular and unitriangular matrices. Our main interest is in the case where S is an idempotent semifield, where we also consider the subsemigroups U Tn(S * ) and Un(S * ) consisting of those matrices of U Tn(S) and Un(S) having all elements on and above the leading diagonal non-zero. Our guiding examples of such S are the 2-element Boolean semiring B and the tropical semiring T. In the first case, Mn(B) is isomorphic to the semigroup of binary relations on an n-element set, and in the second, Mn(T) is the semigroup of n × n tropical matrices.It is well known that the subsemigroup of Mn(F ) consisting of the singular matrices is idempotent generated. We begin consideration of the analogous questions for Mn(S) and its subsemigroups. In particular we show that the idempotent generated subsemigroup of U Tn(T * ) is precisely the semigroup Un(T * ).Il'in has proved that for any semiring R and n > 2, the semigroup Mn(R) is regular if and only if R is a regular ring. We therefore base our investigations for Mn(S) and its subsemigroups on the analogous but weaker concept of being Fountain (formerly, weakly abundant). These notions are determined by the existence and behaviour of idempotent left and right identities for elements, lying in particular equivalence classes. We show that certain subsemigroups of Mn(S), including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We give a detailed study of a family of Fountain semigroups arising in this way that has particularly interesting and unusual properties.

show abstract

Section: 1mentioning

confidence: 99%

Matrix semigroups over semirings

Gould

Johnson

Naz

2019

Int. J. Algebra Comput.

View full text Add to dashboard Cite

show abstract

“…Lemma 4.10. Let R be a division semiring which is also zero-sum-free (see [Hut81]). Consider the following diagram in Set:…”

Section: Weak Invertibilitymentioning

confidence: 99%

Simplicial distributions, convex categories and contextuality

Kharoof¹,

Okay²

2022

Preprint

View full text Add to dashboard Cite

The data of a physical experiment can be represented as a presheaf of probability distributions. A striking feature of quantum theory is that those probability distributions obtained in quantum mechanical experiments do not always admit a joint probability distribution, a celebrated observation due to Bell. Such distributions are called contextual. Simplicial distributions are combinatorial models that extend presheaves of probability distributions by elevating sets of measurements and outcomes to spaces. Contextuality can be defined in this generalized setting. This paper introduces the notion of convex categories to study simplicial distributions from a categorical perspective. Simplicial distributions can be given the structure of a convex monoid, a convex category with a single object, when the outcome space has the structure of a group. We describe contextuality as a monoid-theoretic notion by introducing a weak version of invertibility for monoids. Our main result is that a simplicial distribution is noncontextual if and only if it is weakly invertible. Similarly, strong contextuality and contextual fraction can be characterized in terms of invertibility in monoids. Finally, we show that simplicial homotopy can be used to detect extremal simplicial distributions refining the earlier methods based on Čech cohomology and the cohomology of groups.

show abstract