Modal Kleene algebras are Kleene algebras enriched by forward and backward box and diamond operators. We formalise the symmetries of these operators as Galois connections, complemetarities and dualities. We study their properties in the associated operator algebras and show that the axioms of relation algebra are theorems at the operator level. Modal Kleene algebras provide a unifying semantics...

We provide a structural description of Bruhat order on the set F2n of fixed-pointfree involutions in the symmetric group S2n which yields a combinatorial proof of a combinatorial identity that is an expansion of its rank-generating function. The decomposition is accomplished via a natural poset congruence, which yields a new interpretation and proof of a combinatorial identity that counts the n...

The maximum segment sum problem is to compute, given a list of integers, the largest of the sums of the contiguous segments of that list. This problem specification maps directly onto a cubic-time algorithm; however, there is a very elegant linear-time solution too. The problem is a classic exercise in the mathematics of program construction, illustrating important principles such as calculatio...

let g = (v,e) be a locally finite graph, i.e. a graph in which every vertex has finitely many adjacent vertices. in this paper, we associate a topology to g, called graphic topology of g and we show that it is an alexandroff topology, i.e. a topology in which intersec- tion of every family of open sets is open. then we investigate some properties of this topology. our motivation is to give an e...

This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...

modeling and simulation of processing plants are widely used in industry. construction of a mathematical model for a plant is a time-consuming and error-prone task. in light of extensive advancements in computer science (both hardware and software), computers are becoming a necessary instrument in industrial activities. many software tools for modeling, simulation and optimization of processing...

Recall that a continuous function $fcolon Xto Y$ between Tychonoff spaces is proper if and only if the Stone extension $f^{beta}colon beta Xtobeta Y$ takes remainder to remainder, in the sense that $f^{beta}[beta X-X]subseteq beta Y-Y$. We introduce the notion of ``taking remainder to remainder" to frames, and, using it, we define a frame homomorphism $hcolon Lto M$ to be $beta$-proper, $lambd...

Let G = (V,E) be a locally finite graph, i.e. a graph in which every vertex has finitely many adjacent vertices. In this paper, we associate a topology to G, called graphic topology of G and we show that it is an Alexandroff topology, i.e. a topology in which intersec- tion of every family of open sets is open. Then we investigate some properties of this topology. Our motivation is to give an e...

This paper is an attempt to apply the reasoning principles and calculational style underlying the so-called Bird-Meertens formalism to the design of process calculi, parametrized by a behaviour model. In particular, basically equational and pointfree proofs of process properties are given, relying on the universal characterisation of anamorphisms and therefore avoiding the explicit construction...

In this note we present a new treatment of the pointfree version of the semicontinuous quasi-uniformity based on the new tool of the ring of arbitrary (not necessarily continuous) real-valued functions made available recently by J. Gutiérrez Garćıa, T. Kubiak and J. Picado [Localic real functions: a general setting, Journal of Pure and Applied Algebra 213 (2009) 1064-1074]. The purpose is to sh...