Equational theories of some linear equations are studied. As a consequence, semimodule representations of the corresponding algebras are obtained. Examples are shown on medial and paramedial equations and some of their general-

We provide an expression for the conductor \(c(\varDelta )\) of a semimodule \(\varDelta \) numerical semigroup \(\varGamma with two generators in terms syzygy module and semigroup. In particular, we deduce that difference between is element \), as well formula dual \).

We find sufficient conditions for a subclone C of Burle’s clone and for a subclone of the polynomial clone of a finite semimodule to have the property that the associated C-minor partial order has finite principal ideals. We also prove that for these clones the C-minor partial order is universal for the class of countable partial orders whose principal ideals are finite.

On the basis of the concept of the interval valued intuitionistic fuzzy sets introduced by K. Atanassov, the notion of interval valued intuitionistic fuzzy subsemimodule of a semimodule with respect to t-norm T and s-norm S is given and the characteristic properties are described. The homomorphic image and inverse image are investigated. In particular, by the help of the congruence relations on...

In this note we describe conditions under which, in idempotent functional analysis (see [1–3]), linear operators have integral representations in terms of idempotent integral of V. P. Maslov. We define the notion of nuclear idempotent semimodule and describe idempotent analogs of the classical kernel theorems of L. Schwartz and A. Grothendieck (see, e.g., [4–6]). In [11], for the idempotent sem...

In this paper we propose a semiring-theoretic approach to MV-algebras based on the connection between such algebras and idempotent semirings established in [11] and improved in [4] — such an approach naturally imposing the introduction and study of a suitable corresponding class of semimodules, called MV-semimodules. Besides some basic yet fundamental results of more general interest for semiri...

We define the family of n-gram embeddings from strings over a finite alphabet into the semimodule N . We classify all ∈ N that are valid images of strings under such embeddings, as well as all whose inverse image consists of exactly 1 string (we call such uniquely decodable). We prove that for a fixed alphabet, the set of all strings whose image is uniquely decodable is a regular language. © 20...

We introduce max-plus analogues of basic Euclidian geometry notions: scalar product is replaced by a scalar division, and the associated distance is essentially Hilbert’s projective distance. We introduce an orthogonal projection and prove a Hahn-Banach type theorem: a point can be separated from a semimodule by a hyperplane orthogonal to the direction of projection. We use these results to sep...