We consider properties of residuated lattices with universal quantifier and show that, for a residuated lattice $X$, $(X, forall)$ is a residuated lattice with a quantifier if and only if there is an $m$-relatively complete substructure of $X$. We also show that, for a strong residuated lattice $X$, $bigcap {P_{lambda} ,|,P_{lambda} {rm is an} m{rm -filter} } = {1}$ and hence that any strong re...

The concept of dialgebra provides a platform under which universal algebra and coalgebra are unified in one theory. Other examples of dialgebras include universal multialgebras and partial algebras. In the dialgebraic setting, several relationships between common features of these various theories are clarified and, in many cases, rather similar proofs of closely related results are combined to...

Given a directed graph, there exists a universal operator algebra and universal C-algebra associated to the directed graph. In this paper we give intrinsic constructions of these objects. We provide an explicit construction for the maximal C-algebra of an operator algebra. We also discuss uniqueness of the universal algebras for finite graphs, showing that for finite graphs the graph is an isom...

1 Algebras In this first section we will consider some common features of familiar algebraic structures such as groups, rings, lattices, and boolean algebras to arrive at a definition of a general algebraic structure. Recall that a group G consists of a nonempty set G, along with a binary operation · : G → G, a unary operation −1 : G → G, and a constant 1 G such that • x · (y · z) = (x · y) · z...

Alfred North Whitehead promoted the idea of UNIVERSAL ALGEBRA in his monumental treatise of 1897 [1]. He proposed two candidates for this lofty title, the algebra of Boole and Grassmann’s Algebra of Extension. Boolean algebra has since secured universality status in Set Theory and Symbolic Logic, although only the former is universally known and used by mathematicians. However, Whitehead’s work...

These are notes of a mini-course given at Dennisfest in June 2001. The goal of these notes is to give a self-contained survey of deformation quantization, operad theory, and graph homology. Some new results related to “String Topology” and cacti are announced in Section 2.7.