The intersection graph $mathbb{Int}(A)$ of an $S$-act $A$ over a semigroup $S$ is an undirected simple graph whose vertices are non-trivial subacts of $A$, and two distinct vertices are adjacent if and only if they have a non-empty intersection. In this paper, we study some graph-theoretic properties of $mathbb{Int}(A)$ in connection to some algebraic properties of $A$. It is proved that the fi...

In this paper, we first consider (po-)torsion free and principally weakly (po-)flat $S$-posets, specifically  we discuss when (po-)torsion freeness implies principal weak (po-)flatness. Furthermore, we give a counterexample to show that Theorem 3.22 of Shi is incorrect. Thereby we present a correct version of this theorem. Finally, we characterize pomonoids over which all cyclic $S$-posets are ...

We give new a proof of the general Briançon-Skoda theorem about ideals of holomorphic functions by means of multivariable residue calculus. The method gives new variants of this theorem for products of ideals. Moreover, we obtain a related result for the ideal generated by the the subdeterminants of a matrix-valued generically surjective holomorphic function, generalizing the duality theorem fo...

the minimal prime decomposition for semiprime ideals is defined and studied on z-ideals of c(x). the necessary and sufficient condition for existence of the minimal prime decomposition of a z-ideal / is given, when / satisfies one of the following conditions: (i) / is an intersection of maximal ideals. (ii) i is an intersection of o , s, when x is basically disconnected. (iii) i=o , when x x ha...

This paper defines reduction on derivations in the strict intersection type assignment system of [1], by generalising cut-elimination, and shows a strong normalisation result for this reduction. Using this result, new proofs are given for the approximation theorem and the characterisation of normalisability using intersection types.

This paper defines reduction on derivations in the strict intersection type assignment system of [2], by generalising cut-elimination, and shows a strong normalisation result for this reduction. Using this result, new proofs are given for the approximation theorem and the characterisation of normalisability using intersection types.

We give an alternative proof of the Witten-Di Francesco-ItzyksonZuber theorem —which expresses derivatives of the generating functional of intersection numbers as matrix integrals— using techniques based on diagrammatic calculus and combinatorial relations among intersection numbers on the moduli spaces of algebraic curves. Our techniques extend to a more general interaction potential.

In simply typed λ-calculus with one ground type the following theorem due to Loader holds. (i) Given the full model F over a finite set, the question whether some element f ∈ F is λ-definable is undecidable. In the λ-calculus with intersection types based on countably many atoms, the following is proved by Urzyczyn. (ii) It is undecidable whether a type is inhabited. Both statements are major r...

In this largely expository note we give some homological properties of algebraic maps of complex algebraic varieties which are rather surprising from the topological point of view. These include a generalisation to higher dimension of the invariant cycle theorem for maps to curves. These properties are all corollaries of a recent deep theorem of Deligne, Gabber, Beilinson, and Bernstein which i...