Introduction and Technical Preliminaries - Big Data Integration Theory - page 5

invitation to work at College Park University, MD, USA, on some algebraic prob-
lems in temporal probabilistic logic and databases. Only after 2007, I was again able
to consider these problems of the DB category and to conclude this work. Different
properties of this DB category were presented in a number of previously published
papers, in initial versions, [ 53 - 57 ] as well, and it has been demonstrated that this
category is a weak monoidal topos . The fundamental power view-operator T has
been defined in [ 52 ]. The Kleisli category and the semantics of morphisms in the
DB category, based on the monad (endofunctor T ) have been presented in [ 59 ]. The
semantics for merging and matching database operators based on complete database
lattice, as in [ 7 ], were defined as well and presented in a number of papers cited
above. But in this topic, the new material represents more than 700 percent w.r.t.
previously published research.
In what follows, in this chapter we will present only some basic technical notions
for algebras, database theory and the extensions of the first-order logic language
(FOL) for database theory and category theory that will be used in the rest of this
work. These are very short introductions and more advanced notions can be found
in given references.
This work is not fully self-contained; it needs a good background in Relational
Database theory, Relational algebra and First Order Logic. This very short intro-
duction is enough for the database readers inexperienced in category theory but
interested in understanding the first two parts of this topic (Chaps. 2 through 7 )
where basic properties of the introduced DB category and Categorical semantics
for schema database mappings based on views, with a number of more interesting
applications, are presented.
The third part of this topic is dedicated to more complex categorical analysis of
the (topological) properties of this new base DB category for databases and their
mappings, and it requires a good background in the Universal algebra and Category
theory.

1.2 Introduction to Lattices, Algebras and Intuitionistic Logics

Lattices are the posets (partially ordered sets) such that for all their elements a and
b ,theset

has both a join (lub—least upper bound) and a meet (glb—greatest
lower bound)) with a partial order

{ a,b }

(reflexive, transitive and anti-symmetric).
A bounded lattice has the greatest (top) and least (bottom) element, denoted by
convention as 1 and 0. Finite meets in a poset will be written as 1 ,

≤

∧ and finite joins

as 0 , ∨ .By (W, ≤ , ∧ , ∨ , 0 , 1 ) we denote a bounded lattice iff for every a,b,c ∈ W

the following equations are valid:
1. a ∨ a = a , a ∧ a = a , (idempotency laws)

2. a ∨ b = b ∨ a , a ∧ b = b ∧ a , (commutativity laws)

3. a ∨ (b ∨ c) = (a ∨ b) ∨ c , a ∧ (b ∧ c) = (a ∨ b) ∧ c , (associativity laws)

4. a ∨ 0 = a , a ∧ 1 = a ,

5. a ∨ (b ∧ a) = a , a ∧ (b ∨ a) = a . (absorption laws)