Application of Hohle's Square Roots on Hoop Algebras

TL;DR

This paper explores the application of Hohle's square root concept to hoop algebras, introducing new classes and analyzing their properties, including the formation of square roots on quotient structures and the characterization of good hoop algebras.

Contribution

It defines square roots in hoop algebras, introduces good hoop algebras, and proves that bounded hoop algebras with square roots form a variety.

Findings

Square roots can be defined and studied within hoop algebras.

The formation of square roots on quotient structures is well-behaved.

The class of bounded hoop algebras with square roots is a variety.

Abstract

Square root is a useful tool to study the properties of (ordered) algebraic structures. In this article, we are going to employ this tool to study hoop algebras. To do so, we define square root and make the first attempt to explore the significance properties of this concept in this setting. Then, due to the key role of square roots in obtaining new hoop algebras, we apply them on the filters of hoop algebras, and show that the formation of square roots on quotient structures of hoop algebras by their filters is well-behaved. In addition, a new class of hoop algebras having square roots, so called good hoop algebras, is introduced and its relationships with other classes of ordered algebras such as Boolean algebras and Godel algebras are explored. Several examples are provided as well. Ultimately, it is shown that the class of all bounded hoop algebras with square roots is a variety.