Theory

Before presenting the functionalities of this framework, we need to introduce some important theoretical concepts.

The aim of this library is to propose operations on lattice theory.

Lattice theory was introduced by Birkhoff in his book Trends in LATTICE THEORY

Let’s first introduce the notion of an order relation.

A binary relation \(R\) on a set \(E\) is a set of pairs \((x,y)\), which is a subset of all the pairs \(\in E^2\), We can also write \(xRy\) to say \(x\) has \(R\) relation with \(y\).

We say that \(R\) is an order relation and we note it \(\leq\) if it is:

  • reflexive: \(a \leq a\)

  • transitive: \(a \leq b\) and \(b\leq c\) implies \(a \leq c\)

  • anti-symmetric: \(a \leq b\) and \(b \leq a\) implies \(a = b\)

An ordered set is a pair \((E,\leq)\), with \(E\) being a set and \(\leq\) an order relation on \(E\).

\(a\) is called lower neighbour of \(b\), if \(a < b\) and there is no element \(c\) such as \(a < c < b\), so in this case \(b\) is an upper neighbour of \(a\). (we write: \(a \prec b\)).

Every finite ordered set \((E,\leq)\) can be represented by a line diagram called a Hasse diagram.

Two elements \(x\), \(y\) of an ordered set \((E,\leq)\) are considered comparable if \(x \leq y\) or \(y \leq x\), otherwise they are incomparable.

Let \((E,\leq)\) be an ordered set, and \(X\) a subset of \(E\). A lower bound of \(X\) is an element \(s\) of \(E\) with \(s < x\) for all \(x \in X\). An upper bound of \(X\) is defined dually. If there is a largest element in the set of all lower bounds of \(X\), it is called the infimum of \(X\) and is denoted by \(\inf X\) or \(\bigwedge X\), dually, a least upper bound is called supremum and denoted by \(\sup X\) or \(\bigvee X\).

Infimum and supremum are frequently also called meet and join.

An ordered set \(V = (L,\leq)\) is a lattice if for any two elements \(x, y \in L\), the supremum and the infimum exists.

\(L\) is called a complete lattice, if the supremum \(\bigvee X\) and the infimum \(\bigwedge X\) exist for any subset \(X\) of \(L\). Every complete lattice \(L\) has a largest element, \(\bigvee L\), called the unit element of the lattice, denoted by \(1_L\). Dually, the smallest element \(0_L\) is called the zero element.

Irreducible element: element that can’t be obtained as the join or the meet of any subset of other elements: