Subdirect unions in universal algebra

Subdirect Unions in Universal Algebra

The number of distinct operations (that is, the range of the variable a) may be infinite, but for our main result (Theorem 2), we shall require every n(a) to be finite—that is, it will concern algebras with finitary operations. The concepts of subalgebra, congruence relation on an algebra, homomorphism of one algebra A onto (or into) another algebra with the same operations, and of the direct u...

The algebra of metric betweenness I: Subdirect representation and retraction

We bring together algebraic concepts such as equational class and various concepts from graph theory for developing a structure theory for graphs that promotes a deeper analysis of their metric properties. The basic operators are Cartesian multiplication and gated amalgamation or, alternatively, retraction. Specifically, finite weakly median graphs are known to be decomposable (relative to thes...

Universal Algebra

Universal Algebra in Type Theory

We present a development of Universal Algebra inside Type Theory, formalized using the proof assistant Coq. We define the notion of a signature and of an algebra over a signature. We use setoids, i.e. types endowed with an arbitrary equivalence relation, as carriers for algebras. In this way it is possible to define the quotient of an algebra by a congruence. Standard constructions over algebra...

Universal Dialgebra: Unifying Universal Algebra and Coalgebra

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...