The theory of implicit operations

TL;DR

This paper develops an algebraic theory of implicit operations, focusing on their definition via first order formulas and preservation under homomorphisms within a class of algebras.

Contribution

It introduces a formal algebraic framework for understanding implicit operations defined by first order formulas.

Findings

Defines implicit operations as first order definable and homomorphism-preserved functions.

Establishes foundational properties of implicit operations in algebraic structures.

Abstract

A family of partial functions of a class of algebras $\mathsf{K}$ is said to be an implicit operation of $\mathsf{K}$ when it is defined by a first order formula and it is preserved by homomorphisms. In this work, we develop the theory of implicit operations from an algebraic standpoint.