esialgset

Esialgset is a mathematical concept used to describe a class of subsets of an affine space over a field, defined as the common zero locus of a family of esialg functions. An esialg function is a member of a generalized function algebra that extends ordinary polynomials by incorporating additional, controlled operations while retaining key algebraic properties such as closure under addition and multiplication. The study of esialgsets focuses on the sets of points where all functions in a given family vanish.

In this framework, esialgsets give rise to a natural topology, often called the esialg topology, in which esialgsets are the closed sets. This topology is intended to generalize the familiar Zariski topology of algebraic geometry and to provide a setting in which generalized solution sets can be analyzed using similar geometric and algebraic ideas. Under suitable finiteness or coherence conditions, esialgsets exhibit behavior parallel to that of algebraic sets, including stability under intersections and a correspondence with ideals in the underlying function algebra.

Relation to algebraic sets is central: every algebraic set defined by ordinary polynomials is an esialgset, making esialgsets a broad generalization. Conversely, esialgsets may include sets that cannot be described by polynomials alone, highlighting new kinds of geometric objects arising from the extended function class.

Examples include the zero loci of finite families of esialg polynomials and certain infinite families governed by coherence conditions. Esialgsets are primarily discussed in theoretical contexts, where they serve as a tool for exploring generalized solution sets, model-theoretic properties, and potential extensions of classical algebraic geometry. The concept is used to study how adding generalized operations to the function alphabet affects geometric and topological structure.