We define and study basic properties of ∗-continuous Kleene ω-algebras that involve a ∗-continuous Kleene algebra with a ∗-continuous action on a semimodule and an infinite product operation that is also ∗-continuous. We show that ∗-continuous Kleene ω-algebras give rise to iteration semiring-semimodule pairs. We show how our work can be applied to solve certain energy problems for hybrid systems.

We study the canonical weak distributive law $\delta$ of powerset monad over semimodule for a certain class semirings containing, in particular, positive semifields. For this subclass we characterise as convex closure free set. Using abstract theory laws, compose and monads via $\delta$, obtaining subsets semimodule.

1. In the first theorem of [3], Wiegandt incorrectly states that a semimodule is complete if and only if it is divisible. (See §2 for definitions.) In this paper we show (in §3) that his theorem should read: a semimodule is complete if and only if it is a divisible group. Divisible semimodules which are not necessarily groups are discussed in §4, in connection with Wiegandt's second theorem. It...

The concept of (A,B)-invariant subspace is the fundamental concept of the geometric approach of control design. It has been extended by many authors to that of (A,B)-invariant module or semimodule, for the sake of extending the solution of various control problems to the case of systems over rings or semi rings. In this paper is discussed the use of dynamic feedback control laws for systems ove...

In [1] a generalisation of Formal Concept Analysis was introduced with data mining applications in mind, K-Formal Concept Analysis, where incidences take values in certain kinds of semirings, instead of the standard Boolean carrier set. A fundamental result was missing there, namely the second half of the equivalent of the main theorem of Formal Concept Analysis. In this continuation we introdu...

A short proof of the characterization of idempotent subreducts of semimodules over commutative semirings is presented. It says that an idempotent algebra embeds into a semimodule over a commutative semiring, if and only if it belongs to the variety of Szendrei modes.

This paper presents a query evaluation technique for positive relational algebra queries with aggregates on a representation system for probabilistic data based on the algebraic structures of semiring and semimodule. The core of our evaluation technique is a procedure that compiles semimodule and semiring expressions into so-called decomposition trees, for which the computation of the probabili...

Modes are idempotent and entropic algebras. Although it had been established many years ago that groupoid modes embed as subreducts of semimodules over commutative semirings, the general embeddability question remained open until M. Stronkowski and D. Stanovský’s recent constructions of isolated examples of modes without such an embedding. The current paper now presents a broad class of modes t...