Strict Order

A strict order (in English: strict order) is a relation that is irreflexive, transitive, and asymmetric. Asymmetry means that if a < b, it can never happen that b < a. This order expresses 'strictly smaller' type connections, where an element never relates to itself, but if one element is smaller than another, and that other is smaller than a third, then the first is also smaller than the third.

Formal Definition

Irreflexive: an element never relates to itself.

Transitive: if a relates to b and b to c, then a relates to c.

Asymmetric: if a relates to b, then b does not relate to a.

Examples of Strict Orders

• The "<" relation on natural numbers: irreflexive, transitive, asymmetric.
• The strict subset relation ⊂ on sets: irreflexive, transitive, asymmetric.
• The "precedes in dictionary order" relation on words: irreflexive, transitive, asymmetric.

Counterexamples (Non-Strict Orders)

• The "≤" relation: not irreflexive (every element relates to itself).
• The "friend of" relation among people: not asymmetric (if A is friend of B, B can be friend of A).
• The "divides" relation on natural numbers: not asymmetric (if 2 divides 4 and 4 divides 2? No, but for equals it would, but since reflexive no, wait – divides is antisymmetric but not asymmetric).

Connection to Non-Strict Orders

Every strict order has a 'non-strict' variant (e.g., < instead of ≤), and vice versa. If there is a strict order R, we can create the non-strict order R′ from it such that (a,b) ∈ R′ if and only if a = b or (a,b) ∈ R. This ensures the close connection between the two concepts.

Summary

A strict order is a relation that is irreflexive, transitive, and asymmetric. This type of relation models 'strictly smaller' comparisons. It plays an important role in mathematics and computer science because many structures are built on strict orders.

Practice Exercise