We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and comultiplication laws, thereby providing a generic scheme furnishing combinatorial classes with an algebraic structure. The paper is meant as a gentle introduction to the conc...

In this work we present novel methods of generating fuzzy implications from the given ones. We show that these methods lead to various and much richer algebraic structures on the set of all fuzzy implications than obtained in the literature so far.

Canonical functions are a powerful concept with numerous applications in the study of groups, monoids, and clones on countable structures with Ramsey-type properties. In this short note, we present a proof of the existence of canonical functions in certain sets using topological dynamics, providing a shorter alternative to the original combinatorial argument. We moreover present equivalent alge...

Let A be a ring with identity. The generalized affine transformation monoid Gaff(A) is defined as the set of all transformations on A of the form x → xu + a (for all x ∈ A), where u,a∈ A. We study the algebraic structure of the monoid Gaff(A) on a finite Galois ring A. The following results are obtained: an explicit description of Green’s relations on Gaff(A); and an explicit description of the...

We prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface, and determine the birational structure of certain moduli space of stable bundles on a rational ruled surface.

In his works [R1, R2] David Robbins proposed several interrelated conjectures on the area of the polygons inscribed in a circle as an algebraic function of its sides. Most recently, these conjectures have been established in the course of several independent investigations. In this note we give an informal outline of these developments.

According to Deligne, the cohomology groups of a complex algebraic variety carry a generalized Hodge structure, or, in precise terms, a mixed Hodge structure [2]. The purpose of this paper is to introduce an absract theory of extensions of mixed Hodge structures which has proved useful in the study of low-dimensional varieties [1]. To give the theory meaning, we will give one simple, but illust...

We solve a recent open problem about a new transformation mapping the set of copulas into itself. The obtained mapping is characterized in algebraic terms and some limit results are proved.

4 Partial rankings 11 4.1 Logic for partial rankings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Entailment between partial rankings . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 Correspondence between partial rankings and tableaux . . . . . . . . . . . . . 24 4.4 Duality between partial rankings and sets of tableaux . . . . . . . . . . . . . . 27 4.5 The domai...