On some classes of noncommutative dimonoids
Volodymyr Gavrylkiv
TL;DR
This work analyzes noncommutative dimonoids by exploiting duality and explicit semigroup constructions. It defines the dual dimonoid $(D,\dashv,\vdash)^d$ via $x\dashv^d y = y\vdash x$ and $x\vdash^d y = y\dashv x$, and shows this operation is involutive with important consequences for abelianity, halos, and automorphisms. The author constructs rich families of examples: abelian noncommutative dimonoids from right-commutative semigroups (e.g., $LO_{A\leftarrow D}$ and variants) and nonabelian noncommutative dimonoids with empty halos (e.g., $LOB_D\rbag O_D^{\{a\}}$), as well as rectangular dimonoids arising from rectangular semigroups (e.g., $LO_D\rbag LO_{A\leftarrow D}$ and $LO_D\rbag RO_{A\leftarrow D}$). For each construction, halos and automorphism groups are determined, and the results support isomorphism classifications for small orders and deepen the structure theory of dimonoids and related dialgebras.
Abstract
The present paper is devoted to the study of dimonoids, algebraic structures with two associative binary operations that satisfy a prescribed system of axioms. We investigate the properties of dual dimonoids. In the class of noncommutative dimonoids, we construct a number of abelian, nonabelian, and rectangular dimonoids. The structure of these objects is analyzed, their automorphism groups and halos are computed.