The lattice of alter egos (Q2882399)

The main aim of the paper is to investigate all possible natural dualities for quasivarieties generated by finite algebras (and, more generally, by finite algebraic structures, called here \textit{base structures}). The paper offers new insight into the relationship between the concepts of duality, full duality and strong duality. In particular, it reveals a new understanding of full dualities at the finite level.NEWLINENEWLINEA base structure \(\langle M;F,R\rangle\) is a finite set \(M\) endowed with a family \(F\) of finitary partial operations and a family \(R\) of finitary relations. The authors compare different base structures on the same set \(M\) by the relation \(\sqsubseteq\) of being \textit{a structural reduct}. This is a quasi-order on the set of all base structures on \(M\). By the natural factorization we obtain a doubly algebraic lattice \({\mathcal S}_M\).NEWLINENEWLINEFor a fixed base structure \({\mathbf M}=\langle M;F,R\rangle\), the authors consider the base structures compatible with \({\mathbf M}\). These structures form a complete sublattice \({\mathcal A}_{\mathbf M}\) of the lattice \({\mathcal S}_M\). This sublattice is called the \textit{lattice of alter egos} of \({\mathbf M}\). The basic theory is used to show that: (1) the alter egos that yield a finite-level duality form a principal filter of \({\mathcal A}_{\mathbf M}\) and, if any of these alter egos yield a duality, then they all do; (2) the alter egos that yield a finite-level strong duality form the top element of the lattice \({\mathcal A}_{\mathbf M}\); (3) the alter egos that yield a finite-level full duality form a complete sublattice \({\mathcal F}_{\mathbf M}\) of \({\mathcal A}_{\mathbf M}\), and those that yield a full duality form an increasing subset of \({\mathcal F}_{\mathbf M}\). The authors illustrate their results via several concrete examples. As the main tools they use a new Galois connection for partial operations as well as the concept of entailment. These results might be of an independent interest.