It could be useful to mention the paper 29 #1239 Higgins, Philip J., Algebras with a scheme of operators. Math. Nachr. 27 1963 115--132. (Reviewer: A. Heller) 18.10 which discusses partial algebras at an early date. Ronnie Brown Till Mossakowski wrote:
It is folklore that the method of generators and relations works for any essentially algebraic theory with finitary operators, as well as for some more general ones. "Algebraic" means defined by operators and equational laws (could be many-sorted); "essentially" means that the operators may be partial, with their domains of definition described by finite sets of equations.
I wish I knew of an introductory text describing the techniques at this level of generality, but unfortunately I'm not aware of any - maybe somebody can suggest one. Manes's book "Algebraic Theories" is quite good on the algebraic case.
@BOOK{Reichel, AUTHOR = "Horst Reichel", TITLE ="Initial Computability, Algebraic Specifications and Partial Algebras", PUBLISHER = "Oxford Science Publications", YEAR = 1987}
contains an elementary introduction to essentially algebraic theories; also, the theory of categories is used as an example.
Till Mossakowski
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