The theory and practice of polytypic programming is intimately connected with the initial algebra semantics of datatypes. This is both a blessing and a curse. It is a blessing because the underlying theory is beautiful and well developed. It is a curse because the initial algebra semantics is restricted to so-called regular datatypes. Recent work by R. Bird and L. Meertens [3] on the semantics ...

We show that the set of cluster monomials for the cluster algebra of type D4 contains a basis of the Z-module Z[x1,1, . . . , x3,3]. We also show that the transition matrices relating this cluster basis to the natural and the dual canonical bases are unitriangular and nonnegative. These results support a conjecture of Fomin and Zelevinsky on the equality of the cluster and dual canonical bases....

We discuss a formal many-valued logic called EQlogic which is based on a recently introduced special class of algebras called EQ-algebras. The latter have three basic binary operations (meet, multiplication, fuzzy equality) and a top element and, in a certain sense, generalize residuated lattices. The goal of EQ-logics is to present a possible direction in the development of mathematical logics...

Barrett et al. asked in [W. Barrett et al. Minimum rank of edge subdivisions of graphs. Electronic Journal of Linear Algebra, 18:530–563, 2009.], whether the maximum nullity is equal to the zero forcing number for all complete subdivision graphs. We prove that this equality holds. Furthermore, we compute the value of M(F, G̊) = Z(G̊) by introducing the bridge tree of a connected graph. Since this...

The paper considers an early approach toward a (fuzzy) set theory with a graded membership predicate and a graded equality relation which had been developed by the German mathematician D. Klaua in 1965. In the context of the mathematical fuzzy logic MTL of left-continuous t-norms we discuss some properties of these graded relations. We compare the simultaneous recursive definitions of these rel...

1. I n t r o d u c t i o n The aim of this note is to explain how a well-known result from Moda~ Logic, S~hlqvist's Theorem, can be applied in the theory of Boolean Algebras with Operators to obtain a large class of identities, called Sahlqvist identities, that are preserved under canonical embedding algebras. These identities can be specified as follows. Let a = { fi : i E I } be a set of (nor...

In order to generalize the Kleene theorem from the free monoid to richer algebraic structures, we consider the non deterministic acceptance by a finite automaton of subsets of vertices of a graph. The subsets accepted in such a way are the equational subsets of vertices of the graph in the sense of Mezei and Wright. We introduce the notion of deterministic acceptance by finite automaton. A grap...

where x ∈ Ŝk is minimal such that ν = λ.x −1 satisfies νi < νi+1 for i = 1, 2 . . . k−1 and νi−νk ≥ 1−k−n, and μ = λ.xy. This conjecture is proved by Kazhdan-Lusztig [KL] and Kashiwara-Tanisaki [KT]. The proof relies on an equivalence between the category of finite-dimensional Uǫ(slk)-modules and a category of negative-level representations of the affine algebra ŝlk which are integrable with re...

The sandwiched Rényi \(\alpha \)-divergences of two finite-dimensional quantum states play a distinguished role among the many versions divergences as tight quantifiers trade-off between error probabilities in strong converse domain state discrimination. In this paper, we show same for normal on an injective von Neumann algebra, thereby establishing operational significance these quantities. Mo...