I A more general theory will result if in the place of R we employ an abstract normed ring. s We use the symbols =, ... ... in more than one sense. No confusion need arise as tie context makes clear the meaning of each such symbol. It is worth while to mention here that the relation of equality = for E1 as well as for E2 is not an independent primitive idea; for, an equivalent set of postulates...

Kleene algebra with tests is an extension of Kleene algebra, the algebra of regular expressions, which can be used to reason about programs. We develop a coalgebraic theory of Kleene algebra with Tests, along the lines of the coalgebraic theory of regular expressions based on deterministic automata. Since the known automata-theoretic presentation of Kleene algebra with tests does not lend itsel...

In this paper, the concept of lacunary almost summability of sequences in locally convex spaces has been defined and investigated. It is also proved Kojima-Schur and SilvarmanToeplitz type theorems for lacunary almost conservatively and lacunary almost regularity of the which transform sequences in a Frechet space into a sequence in an other Frechet space. We also stated that the same results h...

In this paper we consider the algebra (A, f, g), where f and g are m-ary and n-ary operations on the set A. We define the concept of regular algebra and we prove that for a regular medial algebra (A, f, g) there exists a commutative semigroup (A,+) such that the operations f and g have linear representations on (A,+). As a corollary, we show that for a regular medial algebra (A,F ) there exists...

We investigate a generalization of Hopf algebra slq (2) by weakening the invertibility of the generator K, i.e. exchanging its invertibility KK = 1 to the regularity KKK = K. This leads to a weak Hopf algebra wslq (2) and a J-weak Hopf algebra vslq (2) which are studied in detail. It is shown that the monoids of group-like elements of wslq (2) and vslq (2) are regular monoids, which supports th...

In an earlier paper, one of the authors presented an algebra for formulating and solving extremal path problems. There are striking similarities between that algebra and the algebra of regular languages, which lead one to consider whether the previous results can be generalized—for instance to path enumeration problems—and whether the algebra of regular languages can itself be profitably used f...

We generalize the construction of multi-tildes in the aim to provide double multitilde operators for regular languages. We show that the underlying algebraic structure involves the action of some operads. An operad is an algebraic structure that mimics the composition of the functions. The involved operads are described in terms of combinatorial objects. These operads are obtained from more pri...

We study a weak variant of omega algebra, where one of the usual star induction axioms is absent, in the context of recursive regular equations. We present abstract conditions for explicitly defining the omega operation and use them for proving an algebraic variant of Arden’s rule for solving such equations. We instantiate these results in concrete models—languages, traces and relations—showing...

We present a general framework for balancing expressions (terms) in form of so called tree straight-line programs. The latter can be seen as circuits over the free term algebra extended by contexts (terms with a hole) and the operations which insert terms/contexts into contexts. It is shown that for every term one can compute in DLOGTIME-uniform TC$^0$ a tree straight-line program of logarithmi...

In this paper we construct a deformation quantization of the algebra of polynomials of an arbitrary (regular and non regular) coadjoint orbit of a compact semisimple Lie group. The deformed algebra is given as a quotient of the enveloping algebra by a suitable ideal.