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Linear order

In mathematics, a total order or linear order is a partial order in which any two elements are comparable.

Version
v1 · 2026-09-28 · History
Domain-specific #
10420
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomain
Order Theory → Mathematics

Core Idea

Linear order is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In mathematics, a total order or linear order is a partial order in which any two elements are comparable. In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X , which satisfies the following for all a, b and c in X. If a \leq b and b \leq c then a \leq c (transitive).

Scope of Application

  • Strict and non-strict total orders. For delimitation purposes, a total order as defined above is sometimes called non-strict order.

  • Examples. If is any set and an injective function from to a totally ordered set then induces a total ordering on by setting if and only if.

  • Orders on the Cartesian product of totally ordered sets. A real function of n real variables defined on a subset of R n defines a strict weak order and a corresponding total preorder on that subset.

  • Documented setting. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used.

  • Strict and non-strict total orders. For each (non-strict) total order \leq there is an associated relation , called the strict total order associated with \leq that can be defined in two equivalent ways.

Clarity

A clear use of Linear order names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, a total order or linear order is a partial order in which any two elements are comparable.

Manages Complexity

Linear order compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—the lexicographical order on the Cartesian product of a family of totally ordered sets, indexed by a well ordered set, is itself a total order.—and the practical consequence—reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders.

Abstract Reasoning

  1. Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: In mathematics, a total order or linear order is a partial order in which any two elements are comparable.
  3. Check operation and conditions. The set of real numbers ordered by the usual "less than or equal to" (≤) or "greater than or equal to" (≥) relations is totally ordered.
  4. Demand recognition evidence.

Knowledge Transfer

Within the home domain. Knowledge about Linear order transfers literally when a new case preserves the same carrier type, relation, and recognition test. For delimitation purposes, a total order as defined above is sometimes called non-strict order. If is any set and an injective function from to a totally ordered set then induces a total ordering on by setting if and only if. Beyond the home domain. No canonical parent is asserted for Linear order.

Relationships to Other Abstractions

Local relationship map for Linear orderParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Linear orderDOMAINPrime abstraction: Order — is a kind ofOrderPRIMEDomain-specific abstraction: Scattered order — is a kind ofScattered orderDOMAIN

Current abstraction Linear order Domain-specific

Parents (1) — more general patterns this builds on

  • Linear order is a kind of Order Prime

    A linear order is an order in which every pair of elements is comparable.

Children (1) — more specific cases that build on this

  • Scattered order Domain-specific is a kind of Linear order

    A scattered order is a linear order whose differentia is containing no nontrivial densely ordered subset.

Hierarchy paths (3) — routes to 3 parentless roots

Neighborhood in Abstraction Space

Linear order sits in a crowded region of the domain-specific corpus (33rd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Algebraic Structures & Order Relations (18 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08