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. Determine the lattices (L 2, ≤), where L 2=L x L.In this context a lattice is a mathematical structure with two binary operators: \/ and /\. Theorem: Prove that every finite lattice L =. If L is a bounded lattice, then for any element a ∈ L, we have the following identities: The set of +ve integer I + under the usual order of ≤ is not a bounded lattice since it has a least element 1 but the greatest element does not exist.The power set P(S) of the set S under the operations of intersection and union is a bounded lattice since ∅ is the least element of P(S) and the set S is the greatest element of P(S).The dual of any statement in a lattice (L,∧ ,∨ ) is defined to be a statement that is obtained by interchanging ∧ an ∨.įor example, the dual of a ∧ (b ∨ a) = a ∨ a isĪ lattice L is called a bounded lattice if it has greatest element 1 and a least element 0. (a) a ∧ ( a ∨ b) = a (b) a ∨ ( a ∧ b) = a Duality: Then L is called a lattice if the following axioms hold where a, b, c are elements in L: Let L be a non-empty set closed under two binary operations called meet and join, denoted by ∧ and ∨.
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