We define the Lukasiewicz Transform as a residuated map and a homomorphism between semimodules over the semiring reducts of an MV-algebra. Then we describe the “ Lukasiewicz Transform Based” ( LTB) algorithm for image processing. We also propose a strong extension of such concepts and results in the framework of quantales and quantale modules. 1 The Lukasiewicz transform and its applications Th...

We associate with every commutative continuous semiring S and alphabet Σ a category whose objects are all sets and a morphism X → Y is determined by a function from X into the semiring of formal series S⟪(Y ⊎Σ) ⟫ of finite words over Y ⊎Σ, an X × Y -matrix over S⟪(Y ⊎Σ) ⟫, and a function from X into the continuous S⟪(Y ⊎Σ) ⟫-semimodule S⟪(Y ⊎Σ) ⟫ of series of ωwords over Y ⊎Σ. When S is also an...

Weighted search is an essential component of many fundamental and useful algorithms. Despite this, it relatively under explored as a computational effect, receiving not nearly much attention either depth- or breadth-first search. This paper explores the algebraic underpinning weighted search, demonstrates how to implement monad transformer. The development first which can be expressed polynomia...

We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices a class of semirings called semirings. The semiring perspective leads necessary sufficient condition for interval [Formula: see text] be subalgebra an lattice, where is dualizing element. also import some results techniques semimodule theory in study this semirings, generalizing about ...

For several semirings S, two weighted finite automata with multiplicities in S are equivalent if and only if they can be connected by a chain of simulations. Such a semiring S is called “proper”. It is known that the Boolean semiring, the semiring of natural numbers, the ring of integers, all finite commutative positively ordered semirings and all fields are proper. The semiring S is Noetherian...

We present a multidimensional optimization problem that is formulated and solved in the tropical mathematics setting. The problem consists in minimizing a nonlinear objective function defined on vectors in a finite-dimensional semimodule over an idempotent semifield by means of a conjugate transposition operator, subject to the constraints in the form of linear vector inequalities. A complete d...

A bialgebra is a structure which is simultaneously an algebra and a coalgebra, such that the algebraic and coalgebraic parts are “compatible”. Bialgebras are normally studied over a field or commutative ring. In this paper, we show how to apply the defining diagrams of algebras, coalgebras, and bialgebras to categories of semimodules and semimodule homomorphisms over a commutative semiring. We ...

We extend the usual definition of coherence, for modules over rings, to partially ordered right modules over a large class of partially ordered rings, called po-rings. In this situation, coherence is equivalent to saying that solution sets of finite systems of inequalities are finitely generated semimodules. Coherence for ordered rings and modules, which we call po-coherence, has the following ...

We determine the points of the epicyclic topos which plays a key role in the geometric encoding of cyclic homology and the lambda operations. We show that the category of points of the epicyclic topos is equivalent to projective geometry in characteristic one over algebraic extensions of the infinite semifield of “max-plus integers” Zmax. An object of this category is a pair (E,K) of a semimodu...

A new multidimensional optimization problem is considered in the tropical mathematics setting. The problem is to minimize a nonlinear function defined on a finite-dimensional semimodule over an idempotent semifield and given by a conjugate transposition operator. A special case of the problem, which arises in just-in-time scheduling, serves as a motivation for the study. To solve the general pr...