Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.

We analyze structure theories of the power-set of ω1 and compare them relative to Cantor’s Continuum Problem. We also compare these theories with the structure theory of the powerset of ω under the assumption of the axiom of definable determinacy.

Hereditarily finite (HF) set theory provides a standard universe of sets, but with no infinite sets. Its utility is demonstrated through a formalisation of the theory of regular languages and finite automata, including the Myhill-Nerode theorem and Brzozowski’s minimisation algorithm. The states of an automaton are HF sets, possibly constructed by product, sum, powerset and similar operations.

we introduce a new concept of general $g$-$eta$-monotone operator generalizing the general $(h,eta)$-monotone operator cite{arvar2, arvar1}, general $h-$ monotone operator cite{xiahuang} in banach spaces, and also generalizing $g$-$eta$-monotone operator cite{zhang}, $(a, eta)$-monotone operator cite{verma2}, $a$-monotone operator cite{verma0}, $(h, eta)$-monotone operator cite{fanghuang}, $h$-...

Multirelations provide a semantic domain for computing systems that involve two dual kinds of nondeterminism. This paper presents relational formalisations of Kleisli, Parikh and Peleg compositions and liftings of multirelations. These liftings are similar to those that arise in the Kleisli category of the powerset monad. We show that Kleisli composition of multirelations is associative, but ne...

In this paper we give an explicit construction of n × n matrices over finite fields which are somewhat rigid, in that if we change at most k entries in each row, its rank remains at least Cn(logq k)/k, where q is the size of the field and C is an absolute constant. Our matrices satify a somewhat stronger property, we which explain and call “strong rigidity.” We introduce and briefly discuss str...

ABSTRACT: In this paper, a complete construction of quaternary complex Hadamard matrices of order 2 n is obtained using the method of Sylvester construction and Williamson construction. Williamson construction has been generalized to obtain any kind of Hadamard matrices (Complex or Real Numbers). Non-equivalent family of Hadamard Matrices can be obtained using the Generalized Williamson constru...

From the assumption that there is a measurable cardinal κ with o(κ) = κ, we produce a model in which for all x ⊆ אω, HODx does not contain the powerset of אω. We also prove that this assertion requires large cardinals.

Functors which are determined, up to natural isomorphism, by their values on objects, are called DVO (Defined by Values on Objects). We focus on the collection of polynomial functors on a category of sets (classes), and we give a characterization theorem of the DVO functors over such collection of functors. Moreover, we show that the (κ-bounded) powerset functor is not DVO.