Clone Theory and Application

We introduce genads, a minimal equational structure for substitution and variable binding. Locally finite genads are equivalent to abstract clones, Lawvere theories, and locally finite clones.

We develop a comprehensive algebraic theory of lambda calculus through the concept of lambda clones--algebraic structures combining clone theory with lambda abstraction.

We develop the theory of gnets (generalized nets), providing a unified framework for abstract substitution theories and their relationship to clone theory and categorical semantics.

We introduce nameless quantifier algebras, an algebraic framework for first-order logic that eliminates explicit variable names in favor of de Bruijn-style indices. We prove that the category of nameless quantifier algebras is equivalent to the category of locally finite cylindric algebras.