We begin this paper by noting that, in a 1969 paper in the Transactions, M.C.McCord introduced a construction that can be interpreted as a model for the categorical tensor product of a based space and a topological abelian group. This can be adapted to Segal’s very special Γ–spaces indeed this is roughly what Segal did and then to a more modern situation: K ⊗ R where K is a based space and R is...

For a quantaloid Q, considered as a bicategory, Walters introduced categories enriched in Q. Here we extend the study of monad-quantale-enriched categories of the past fifteen years by introducing monad-quantaloid-enriched categories. We do so by making lax distributive laws of a monad T over the discrete presheaf monad of the small quantaloid Q the primary data of the theory, rather than the l...

We provide algebraic semantics together with a sound and complete sequent calculus for information update due to epistemic actions. This semantics is flexible enough to accommodate incomplete as well as wrong information e.g. deceit. We give a purely algebraic treatment of the muddy children puzzle, which moreover extends to situations where the children are allowed to lie and cheat. Epistemic ...

We study the differential equation $$\frac{\partial G}{\partial {{\bar{z}}}}=g$$ with an unbounded Banach-valued Bochner measurable function g on open unit disk $${\mathbb {D}}\subset {{\mathbb {C}}}$$ . prove that under some conditions growth and essential support of such has a bounded solution given by continuous linear operator. The obtained results are applicable to corona problem for algeb...

A notion of reticulation which provides topological properties on algebras has introduced on commutative rings in 1980 by Simmons in [5]. For a given commutative ring A, a pair (L, λ) of a bounded distributive lattice and a mapping λ : A → L satisfying some conditions is called a reticulation on A, and the map λ gives a homeomorphism between the topological space Spec(A) consisting of prime fil...

An additive category in which each object has a Krull-Remak-Schmidt decomposition—that is, finite direct sum decomposition consisting of objects with local endomorphism rings—is known as Krull-Schmidt category. A Hom-finite is an for there commutative unital ring k, such that Hom-set length k-module. The aim this note to provide proof Krull-Schmidt, if and only it split idempotents, indecomposa...

We introduce basic notions and results about relation liftings on categories enriched in a commutative quantale. We derive two necessary and sufficient conditions for a 2-functor T to admit a functorial relation lifting: one is the existence of a distributive law of T over the “powerset monad” on categories, one is the preservation by T of “exactness” of certain squares. Both characterisations ...

Let K be any unital commutative Q-algebra and W any non-empty subset of N. Let z = (z1, . . . , zn) be commutative or noncommutative free variables and t a formal central parameter. Let D〈〈z〉〉 (α ≥ 1) be the unital algebra generated by the differential operators ofK〈〈z〉〉 which increase the degree in z by at least α− 1 and A [α] t 〈〈z〉〉 the group of automorphisms Ft(z) = z−Ht(z) of K[[t]]〈〈z〉〉 w...

If R is a commutative unital ring and M R-module, then each element of EndR(M) determines left EndR(M)[X]-module structure on EndR(M), where the R-algebra endomorphisms EndR(M)[X]=EndR(M)⊗RR[X]. These structures provide very short proof Cayley-Hamilton theorem, which may be viewed as reformulation in Algebra by Serge Lang. Some generalisations theorem can easily proved using proposed method.