The recent development of abstract algebraic logic has led to a reconsideration of the universal algebraic theory of ordered algebras from a new perspective. The familiar equational logic of Birkhoff can be naturally viewed as one of the deductive systems that constitute the main object of study in abstract algebraic logic; technically it is a deductive system of dimension two. Large parts of u...

Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained precedence. A generation later, the definition of monad began to appear extensively in theoretic...

In this chapter we review a category theoretic approach to ontology engineering. Using ideas from universal algebra, we carefully distinguish presentations of ontologies from the ontologies themselves. This leads to a precise notion of “view”, and views can be used both to create new ontologies incorporating existing ontologies while recognising their common classes and relations, and to develo...

Pure terms (without constants) over a signature of many sorted universal algebra and terms with constants from algebra are introduced. Facts on evaluation of a term in some valuation are proved.] provide the notation and terminology for this paper. 1. Terms over a Signature and over an Algebra Let I be a non empty set, let X be a non-empty many sorted set indexed by I, and let i be an element o...

Let I be a non empty set, let X be a non-empty many sorted set indexed by I, and let i be an element of I. Note that X(i) is non empty. In the sequel S will be a non void non empty many sorted signature and V will be a non-empty many sorted set indexed by the carrier of S. Let us consider S, V . The functor S -Terms(V ) yielding a non empty subset of FinTrees(the carrier of DTConMSA(V )) is def...

1 The Basics 3 1.1 Graphs and Digraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Graph Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 The H-colouring Problem and Variants . . . . . . . . . . . . . . . . . . . . . 8 1.4 Cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Polymorphisms . . . . . ....

The aim of this paper is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λ-theories (= equational extensions of untyped λ-calculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to several questions raised in these years as well as several independent results, the state of the art ...