SOME ORDER DUALITIES IN LOGIC, GAMES AND CHOICES

We first present the concept of duality appearing in order theory, i.e. the notions of dual isomorphism and of Galois connection. Then, we describe two fundamental dualities, the duality extension/intention associated with a binary relation between two sets, and the duality between implicational systems and closure systems. Finally, we present two "concrete" dualities occuring in social choice and in choice functions theories.

A CONVERSE OF DUAL TELLEGEN’S THEOREM

Let [Formula: see text] and [Formula: see text] be directed networks having the same number of branches labeled correspondingly. It is proved that if [Formula: see text] for all vectors of corresponding branch voltages [Formula: see text] and currents [Formula: see text] satisfying Kirchhoff’s voltage and current laws in every loop and cutset of [Formula: see text] and [Formula: see text] then the imposed correspondence of branches is the dual isomorphism of networks, so that in particular [Formula: see text] and [Formula: see text] are both planar. This establishes the converse to the dual of Tellegen’s famous theorem.

Inverse semigroups with isomorphic partial automorphism semigroups

It is shown that a so-called shortly connected combinatorial inverse semigroup is strongly lattice-determined “modulo semilattices”. One of the consequences of this theorem is the known fact that a simple inverse semigroup with modular lattice of full inverse subsemigroups is strongly lattice-determined [7]. The partial automorphism semigroup of an inverse semigroup S consists of all isomorphisms between inverse subsemigroups of S. It is proved that if S is a shortly connected combinatorial inverse semigroup, T an inverse semigroup and the partial automorphism semigroups of S and T are isomorphic, then either S and T are isomorphic or they are dually isomorphic chains (with respect to the natural partial order); moreover, any isomorphism between the partial automorphism semigroups of S and T is induced either by an isomorphism or, if S and T are dually isomorphic chains, by a dual isomorphism between S and T. Counter-examples are constructed to demonstrate that the assumptions about S being shortly connected and combinatorial are essential.