Simultaneous Substitution Algebras
Zichen Peng
Abstract:
In this thesis we introduce simultaneous substitution algebras as an abstraction of simultaneous substitution operations on terms and on functions. The class of simultaneous substitution algebras is defined by a set of equations, and we prove that the equational theory generated by this set is decidable and complete with the class of term simultaneous substitution algebras and of polynomial simultaneous substitution algebras. We also prove that each simultaneous substitution algebra can be represented as a quotient of a function simultaneous substitution algebra, and each locally finite-dimensional one can be represented as a polynomial simultaneous substitution algebra. Relevant results in singular substitution algebras can be derived from the results in this thesis.