Algebras of distributions for binary semi-isolating formulas of a complete theory

TL;DR

This paper introduces a new algebraic framework for binary semi-isolating formulas in complete theories, extending previous algebraic models and characterizing their structure as monoids with partial orders and set-theoretic operations.

Contribution

It defines a class of algebras for binary semi-isolating formulas, generalizing existing models of isolating formulas, and characterizes their algebraic structure and properties.

Findings

Binary semi-isolating formulas form monoids with partial orders.

The algebraic structure includes set-theoretic operations and composition.

The class of these algebraic structures is fully described.

Abstract

We define a class of algebras describing links of binary semi-isolating formulas on a set of realizations for a family of 1-types of a complete theory. These algebras include algebras of isolating formulas considered before. We prove that a set of labels for binary semi-isolating formulas on a set of realizations for a 1-type forms a monoid of a special form with a partial order inducing ranks for labels, with set-theoretic operations, and with a composition. We describe the class of these structures. A description of the class of structures relative to families of 1-types is given.