The problem of characterization of the structure of MTL-algebras, which form an equivalent algebraic semantics for Monoidal T-norm Based Logic (see [9]) in the sense of Blok and Pigozzi (see [3]), is still far from being solved. Since MTL-algebras are in fact subdirect products of chains, it suffices to investigate only the structure of MTL-chains if we want to characterize the structure of MTL...

We presents the algebraic theories over an arbitrary monoid, main properties and calculus rules. Ordered, rationaly closed and ω-continuous theories on one hand and matrix and complete matrix theories on the other hand are the presentation main subjects. Some examples comming from algebra and computer science finish this paper.

Following the results obtained in Preord and in Cat, we characterize the effective étale-descent morphisms inM -Ord, the category ofM -ordered sets for a given monoid M . Furthermore we show that in M -Ord every effective descent morphism is effective for étale-descent (while the converse is false), and we generalize it to a more general context of relational algebras.

We introduce two frameworks for constraint solving where classical CSPs, fuzzy CSPs, weighted CSPs, partial constraint satisfaction , and others can be easily cast. One is based on a semiring, and the other one on a totally ordered commutative monoid. We then compare the two approaches and we discuss the relationship between them.

Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of lattice-ordered groups, ideal lattices of rings, linear logic and multi-valued logic. Our exposition aims to cover...

Let I , H , S , P , Pf be the usual operators on classes of algebras of the same type (Pf for filtered products). The partially ordered monoid generated by the operators H , S , P with respect to composition of operators, I as an identity element, and a natural ordering between operators is described by Pigozzi (Algebra Universalis 2 (1972), 346–353). Let us denote by M = 〈H,S, P 〉 and by Mf = ...

Endowing a lower-bounded partially ordered set with a total addition or with a total difference operation leads to the basic notion of a NAM, that is, a naturally ordered abelian monoid, or a BCK-algebra, respectively. BL-algebras may be alteratively viewed as certain NAMs or certain BCK-algebras. We characterize the appropriate subclasses by making use of those properties which have been so fa...

Let C be a finite alphabet, and C* the free monoid generated by Z. Let x, y E C* and x = uyv for U, v E C*. Then y is a factor of x. Moreover if u = 1, then y is a prefix of x, and if u = 1, then y is a suffix of x. Here 2. is the null word. One of the typical string matching problems over C* is the following: given a text string XEX* and a pattern string VEC*, decide whether or not y is a fact...

An interesting generalization of hoop-algebras and commutative residuated lattices is the concept quasi-ordered systems (shortly QRS) introduced in 2018 by Bonzio Chajda. Quasi-ordered system an integral monoid with two internal binary operations interconnected a residuation connection. This specificity reason for complexity this algebraic structure existence significant number substructures it...

A skew generalized power series ring R[[S, ω]] consists of all functions from a strictly ordered monoid S to a ring R whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by convolution twisted by an action ω of the monoid S on the ring R. Special cases of the skew generalized power series ring construction are...