We show that the class of locally finite varieties omitting type 1 has the following properties. This class (1) is definable by an idempotent, linear, strong Mal’cev condition in a language with one 4-ary function symbol. (2) is not definable by an idempotent, linear, strong Mal’cev condition in a language with only one function symbol of arity strictly less than 4. (3) is definable by an idemp...

An approach to schedule development in project management is developed within the framework of idempotent algebra. The approach offers a way to represent precedence relationships among activities in projects as linear vector equations in terms of an idempotent semiring. As a result, many issues in project scheduling reduce to solving computational problems in the idempotent algebra setting, inc...

Optimization problems are considered in the framework of tropical algebra to minimize and maximize a nonlinear objective function defined on vectors over an idempotent semifield, and calculated using multiplicative conjugate transposition. To find the minimum of the function, we first obtain a partial solution, which explicitly represents a subset of solution vectors. We characterize all soluti...

Linear vector equations and inequalities are considered defined in terms of idempotent mathematics. To solve the equations, we apply an approach that is based on the analysis of distances between vectors in idempotent vector spaces. The approach reduces the solution of the equation to that of an optimization problem in the idempotent algebra setting. Based on the approach, existence and uniquen...

This is a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are examined. A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented. Concrete applications to optimizat...

The set of all idempotent probability measures (Maslov measures) on a compact Hausdorff space endowed with the weak* topology determines is functorial on the category Comp of compact Hausdorff spaces. We prove that the obtained functor is normal in the sense of E. Shchepin. Also, this functor is the functorial part of a monad on Comp. We prove that the idempotent probability measure monad conta...

Considering measure theory in which the semifield of positive real numbers is replaced by an idempotent semiring leads to the notion of idempotent measure introduced by Maslov. Then, idempotent measures or integrals with density correspond to supremums of functions for the partial order relation induced by the idempotent structure. In this paper, we give conditions under which an idempotent mea...

A unary operator f is idempotent if the equation f(x) = f(f(x)) holds. On the other end, an element a of an algebra is said to be an idempotent for a binary operator if a = a a. This paper presents a rule format for Structural Operational Semantics that guarantees that a unary operator be idempotent modulo bisimilarity. The proposed rule format relies on a companion one ensuring that certain te...

We generalize the word problem for groups, the formal language of all words in the generators that represent the identity, to inverse monoids. In particular, we introduce the idempotent problem, the formal language of all words representing idempotents, and investigate how the properties of an inverse monoid aer related to the formal language properties of its idempotent problem. In particular,...

Idempotent filters produce a root signal in a single filter pass, i.e., the filter output is invariant to further filterings with the same filter. In this paper median based idempotent filter structures are introduced. Two approaches to generate these filters are studied: weighted median filters and median filter cascades. Two subclasses of n-dimensional idempotent weighted median filters, call...