Seselja, Branimir, Kolokvij september 2015

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The topic of the talk are algebraic systems in which algebras (like groups, rings etc.) are by special functions related to ordered-theoretic structures (e.g., posets, lattices).

We start with a particular algebraic setting developed in the 20th century. Namely, adopting a logic approach to sheaves by M.P. Fourman and D.S. Scott (1977), which was later developed in topology by U. Hoehle (2007), we investigate so-called Ω-sets. If Ω is a complete lattice, then an Ω-poset is a structure (M,R,E), where M is a nonempty set, and R, E are mappings from M2 to Ω such that R(x,y) \wedge R(y,z)\leq R(x,z), and E(x,y)=R(x,y) \wedge R(y,x). Using the cut technique, we define an Ω-lattice as an Ω-poset in which particular quotients with respect to cuts of E are classical lattices.

On the other hand, we start with a bi-groupoid M equipped with a map E from M2 to Ω so that (M,E) is an Ω-set (in the sense of Fourman and Scott). If this structure fullfils particular lattice theoretic formulas, we obtain an Ω-lattice as an algebra. We investigate properties of this structure, including some identities which should be satisfied by the bi-groupoid. We prove that under a particularly defined map R from M2 to Ω, (M,R) is an Ω-poset, moreover it is an Ω-lattice as an Ω-poset.

For an Ω-lattice as an Ω-poset, assuming Axiom of Choice, we define operations on M, so that this structure becomes an Ω-lattice as an algebra.

An analogue approach is used to introduce Ω-groups.

Some general properties of the above structures and concrete examples are also presented.

As an application, we present a construction of an Ω-(lattice valued) relation arising from a collection of objects, individuals or data, organized in particular groups. We are motivated by managing databases or by analyzing social networks and these groups are supposed to have properties usually appearing in some real systems. Mathematically, we start with an arbitrary collection of subsets and we want them to be cuts of blocks of a lattice valued relation. We develop an algebraic method, which enable an algorithm for the construction of a lattice and a lattice valued relation, such that the cuts of its blocks are precisely the sets of the given collection. Applications are explained through a concrete example.

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