Algebraic Properties of the Category of Involutive m-Semilattices and Its Limits
Shaohui Liang · Symmetry · 2025
An involutive m-semilattice is a kind of algebraic structure with symmetry. Symmetry is reflected from partial-order relations to algebraic operations and even categorical properties. In this study, firstly, the concepts of the nucleus and congruence in involutive m-semilattices are introduced, and their interrelationships are discussed. On this basis, the concrete structure of a coequalizer in the category of involutive m-semilattices is obtained. We introduce the definition of free involutive m-semilattices, and the concrete structure of involutive m-semilattices is discussed. In addition, It is shown that the category of involutive m-semilattices is algebraic. Secondly, the colimit in the category of involutive m-semilattices is shown to be a very difficult problem. We obtain the concrete structure of the colimit for a full subcategory of the category of involutive m-semilattices. Thirdly, we introduce the definition of an inverse system in the category of involutive m-semilattices and give the concrete structure of the inverse limit of an inverse system. We establish the concept of a mapping between two inverse systems. The properties of inverse limits are discussed. Finally, we study the direct limit of the category of involutive m-semilattices and give its concrete structure.