Finite Orbits of Language Operations

We consider a set of natural operations on languages, and prove that the orbit of any language L under the monoid generated by this set is finite and bounded, independently of L. This generalizes previous results about complement, Kleene closure, and positive closure

Finite Orbits of Language Operations

We consider a set of natural operations on languages, and prove that the orbit of any language L under the monoid generated by this set is finite and bounded, independently of L. This generalizes previous results about complement, Kleene closure, and positive closure

A uniform approach to normality for topological spaces

[EN] $(\lambda, \mu)$-regularity and $(\lambda, \mu)$-normality are defined for generalized topological spaces. Several variants of normality existing in the literature turn out to be particular cases of $(\lambda, \mu)$-normality. Uryshon's lemma and Titze's extension theorem are discussed in the light of ($\lambda, \mu$)-normality. Gupta, A.; Sarma, RD. (2016). A uniform approach to normality for topological spaces. Applied General Topology. 17(1):7-16. doi:10.4995/agt.2016.3919.SWORD71617

Boundary-Border Extensions of the Kuratowski Monoid

The Kuratowski monoid $\mathbf{K}$ is generated under operator composition by closure and complement in a nonempty topological space. It satisfies $2\leq|\mathbf{K}|\leq14$. The Gaida-Eremenko (or GE) monoid $\mathbf{KF}$ extends $\mathbf{K}$ by adding the boundary operator. It satisfies $4\leq|\mathbf{KF}|\leq34$. We show that when $|\mathbf{K}|<14$ the GE monoid is determined by $\mathbf{K}$. When $|\mathbf{K}|=14$ if the interior of the boundary of every subset is clopen, then $|\mathbf{KF}|=28$. This defines a new type of topological space we call $Kuratowski\ disconnected$. Otherwise $|\mathbf{KF}|=34$. When applied to an arbitrary subset the GE monoid collapses in one of $70$ possible ways. We investigate how these collapses and $\mathbf{KF}$ interdepend, settling two questions raised by Gardner and Jackson. Computer experimentation played a key role in our research.Comment: 48 pages, 9 figure

Modele otoczeniowe i topologiczne dla klasycznych i intuicjonistycznych logik modalnych

We may speak about syntax. From this point of view any logic can be considered as as the set of axioms and rules. Here we are interested in formal proofs and deduction systems. Second, we can also think about semantics, namely, about some models in which it is possible to de ne the notions of truth and falsity. As for the logical calculi, we are working with propositional logics. Thus, we are not so much interested in quanti ers. Our logics are non-classical. Of course, there are many kinds of non-classical logic and many reasons for which certain system can be considered as nonclassical. In our case, there are two main ways which are notoriously combined. On the one hand, we are interested in intuitionistic, superintuitionistic and subintuitionistic systems. This means that we narrow down the set of axioms and rules of classical logic. On the other hand, we use modal operators to de ne and analyse the ideas of necessity and possibility. As a result, we often obtain classical and intuitionistic modal logics. Our semantic models are mostly neighborhood, topological and relational. These three approaches are also combined. For this reason, we may speak about bi-relational and relational-neighborhood structures. Moreover, we go beyond the standard notion of topology in order to study its various generalizations. Finally, our aim is to investigate several non-classical calculi using all the tools mentioned above. We are interested in the issues of completeness (axiomatization), nite model property, bisimulation and decidability. Moreover, we analyse some purely topological properties of the structures in question. The philosophical aspect is also important