Algebraic closures and their variations

TL;DR

This paper explores various forms of algebraic closures in first-order structures, examining their properties, differences, and characteristics across different theories and set bounds, including algebraic closure operators and their algebraic structures.

Contribution

It introduces and analyzes new characteristics of algebraic closures and their variations, as well as algebraic closure operators and their semilattice and lattice structures.

Findings

Characteristics for algebraic closure possibilities are described.

Differences between definable and algebraic closures are analyzed.

Structures of algebraic closure operators are characterized.

Abstract

We study possibilities for algebraic closures, differences between definable and algebraic closures in first-order structures, and variations of these closures with respect to the bounds of cardinalities of definable sets and given sets of formulae. Characteristics for these possibilities and differences are introduced and described. These characteristics are studied for some natural classes of theories. Besides algebraic closure operators with respect to sets of formulae are introduced and studied. Semilattices and lattices for families of these operators are introduced and characteristics of these structures are described.