Weighted automata are a generalisation of non-deterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for non-deterministic automata, their behaviours can be expressed in terms of either (weighted) bisimilarity or (weighted) language equivalence. Coalgebras provide a categorical framework for the uniform study of state-based systems and their behaviours. In this work, we show that coalgebras can suitably model weighted automata in two different ways: coalgebras on Set (the category of sets and functions) characterise weighted bisimilarity, while coalgebras on Vect (the category of vector spaces and linear maps) characterise weighted language equivalence. Relying on the second characterisation, we show three different procedures for computing weighted language equivalence. The first one consists in a generalisation of the usual partition refinement algorithm for ordinary automata. The second one is the backward version of the first one. The third procedure relies on a syntactic representation of rational weighted languages.

/pdf/a-coalgebraic-perspective-on-linear-weighted-automata-5d2g1xbl7y.pdf

A coalgebraic perspective on linear weighted automata

Completely 0-simple semigroups of quotients☆

Simple Commutative Semirings

In this paper, we consider from two different perspectives the concept of "Morita equivalence" in a (non-additive) semiring setting, as well as its application to homological characterization of semirings. Among other results, we present an analog of the Eilenberg–Watts theorem for module categories in the semimodule setting and give various homological characterizations of semisimple and subtractive semirings. We also solve Problem 3.9 in [Y. Katsov, On flat semimodules over semirings, Algebra Universalis51 (2004) 287–299] for the class of additively regular semisimple semirings, showing that for semimodules over semirings of this class the concepts of "mono-flatness" and "flatness" coincide.

Morita equivalence and homological characterization of semirings

A closed description of observation processes must necessarily include the observer, too. In order to find a basis for such a unified description, an operator algebra is developed which enables a formal description of at least a significant majority of cognitive processes. It is found that this operator algebra, which is a noncommutative semiring of a type already known in literature, has astonishing similarities with the usual operator algebras in quantum theory. Combined with a method for the formal treatment of perspective notions, the representation scheme proposed here may open a chance for a unified description of a process itself together with the relevant cognitive processes on the observer's side. By the description of parallel-processing systems on the same basis an operational definition of 'internal time' becomes possible.

http://directory.umm.ac.id/Data%20Elmu/jurnal/B/Biosystems/Vol54.Issue3.2000/1933.pdf

Towards a closed description of observation processes.

This chapter discusses semirings and semifields. Among many concepts of universal algebras, semirings are those that originate from rings, by cancelling the assumption that ( R , +) has to be a group. Depending on how much other ring-like properties are also cancelled or added, various different concepts of semirings (S, +, ·) have been considered. Semirings with different properties have become important in theoretical computer science. A universal algebra S = (S, +, ·) with a nonempty set S and two binary operations, written as addition and multiplication, is called a “semiring” if (S, +) and (S,·) are arbitrary semigroups, such that a ( b + c ) = ab + ac and ( b + c ) a = ba + ca hold for all a, b, c ∈ S . In particular, ( S , +,·) is called a “proper semiring” if ( S , +) is not a group. Each distributive lattice is a semiring, clearly commutative and idempotent with respect to both operations. It has a zero or an identity if it is bounded from below or above, respectively.

Semirings and Semifields

During the last two decades, a good deal of work has been conducted on S-sets and semigroups of quotients, establishing several concepts and results which are more or less analogous to the classicalas well as ring-theoretical ones. Clearly there are different (and sometimes, we feel, too restrictive) assumptions in these papers, in particular on the semigroups S and on the S-sets under consideration, and various notations need to be unified. Generalizing diverging dispositions, we try to give a self-contained survey of the most fundamental parts (and other considerable parts) of these investigations, and add some new results. A forthcoming paper will deal with S-semimodules over semirings S, including some relationships with rings and modules.

S-sets and semigroups of quotients

In this paper we deal with some basic properties of seminearrings (S,+,·), in particular with respect to multiplicative cancellativity (§2) and to seminearfields (§3, §4). All seminearrings of order 2 are listed in §5. In general, we do not assume that (S,+) is associative. Moreover, we present most of our results as consequences of corresponding statements on semigroups (S,·). Clearly, all results apply to semirings and nearrings, particularly to those with associative addition, and some of them will be new also in this context.

Seminearrings, seminearfields and their semigroup-theoretical background

In this paper we obtain characterizations of classes of semirings by P-injective and projective right R-semimodules. We prove that a semiring R is von Neumann regular if and only if each cyclic right R-semimodule is P-injective. Moreover, a commutative semiring R whose principal ideals are k-closed is von Neumann regular if and only if every simple R-semimodule is PP-injective. We also examine some properties of right PP-semirings, that is, semirings all of whose principal right ideals are projective. It is shown that R is a right PP-semiring if and only if the endomorphism semiring of every cyclic projective right R-semimodule is right PP.

Hanns Joachim Weinert

Papers

Semirings and Semifields

S-sets and semigroups of quotients

Seminearrings, seminearfields and their semigroup-theoretical background

Characterizations of semirings by p-injective and projective semimodules

A kurosh-amitsur radical theory for proper semifields