![]() ![]() In the mathematical discipline of order theory, a complemented lattice is a bounded lattice in which every element a has a complement, i.e. any finite topological space has a lattice of sets as its family of open sets.the Boolean lattice defined from the family of all subsets of a finite set has this property.Warning: different sources somtimes reverese the way that /\ and \/ are used so I am not sure which is the best standard to use? Birkhoff's theoremĪll distributive lattices can be represented by sets as follows: distributive lattice Join: where two branches meet beneath we have a join \/ We can represent a poset as a diagram: meet: where two branches meet above we have a meet /\ In join we go downwards in the diagram, from each of the nodes representing the operands, to find the first common node where they first join. In meet we go upwards in the diagram, from each of the nodes representing the operands, to find the first common node where they first meet. We can convert the above 'is-a-factor-of' table into an 'is-direct-factor-of' table, So since 4 is a factor of 8 we don't need to include the factors of 4 as direct factors of 8: Is-direct-factor-ofįrom this diagram we can create two operations called meet and join. So the function 'is-a-factor-of' has a type signature of (int,int) -> boolean The entry in the table is a boolean value where '' represents true and blank represents false depending on whether row is a factor of column: Is-a-factor-of The following table the columns represents an integer we are trying to factorise and the row represents the the integer we are trying to factorise by. identityĪs an example of a lattice we will look at factorising an integer, in this case 120. There are also distributive laws which may, or may not, apply for a particular lattice. By imposing additional conditions (in form of suitable identities) on these operations, one can then derive the underlying partial order exclusively from such algebraic structures.Īn important algebra is a lattice, it has two operations: It turns out that in many cases it is possible to characterize completeness solely by considering appropriate algebraic structures in the sense of universal algebra, which are equipped with operations like \/ or /\. The presence of certain completeness conditions allows us to regard the formation of certain suprema and infima as total operations of an ordered set. If a poset/lattice has completeness properties then it can be described as an algebraic structure. The relational product r o s of two binary relations r,s on A is given by r o s iif for some c: r s. ![]() If a lattice represents a completely ordered set (a>b or a r iff r This has some similarities to a graph structure however the lines in a lattice structure have a notion of 'ordering' and there will be a highest node at the top and a lowest node at the bottom.Ī big application of mathematical lattices is when factoring is involved somehow. This lattice structure can be visualised as a diagram with nodes and lines between them, this diagram is called a Hasse diagram. We can define them using algebraic identities.We can define them using the concept of 'order', that is, as a partially ordered set (poset).So we can approach lattices in different ways: These branches of mathematics tend to use different names and notations for corresponding concepts as follows: Lattice The lattice structure arises in algebras associated with various branches of mathematics including logic, sets and orders. In this context a lattice is a mathematical structure with two binary operators: \/ and /\. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |