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Linearly ordered group

In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant.

Version
v1 · 2026-09-28 · History
Domain-specific #
10426
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Abstract Algebra, Ordered Algebraic Structures → Mathematics

Core Idea

Linearly ordered group is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant. In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant. left-ordered group if ≤ is left-invariant, that is a ≤ b implies ca ≤ cb for all a, b, c in G,. right-ordered group if ≤ is right-invariant, that is a ≤ b.

Scope of Application

  • Further definitions. The set of positive elements in an ordered group is called the positive cone, it is often denoted with G+ ; the slightly different notation G^+ is used for the positive cone.

  • Further definitions. In this section, \le is a left-invariant order on a group G with identity element e .

  • Further definitions. All that is said applies to right-invariant orders with the obvious modifications.

  • Further definitions. Note that \le being left-invariant is equivalent to the order \le' defined by g \le' h if and only if h^{-1} \le g^{-1} being right-invariant.

  • Further definitions. In particular, a group being left-orderable is the same as it being right-orderable.

Clarity

A clear use of Linearly ordered group names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant.

Manages Complexity

Linearly ordered group compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—the positive cone G+ characterises the order \le ; indeed, by left-invariance we see that g \le h if and only if g^{-1} h \in G+ .—and the practical consequence—in these cases, one may classify a group by its rank: which is related to the order type.

Abstract Reasoning

  1. Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant.
  3. Check operation and conditions. The order \leP associated with P is defined by g \leP h \Leftrightarrow g^{-1} h \in P ; the first condition amounts to left-invariance and the second to the order being well-defined and total. 4.

Knowledge Transfer

Within the home domain. Knowledge about Linearly ordered group transfers literally when a new case preserves the same carrier type, relation, and recognition test. The set of positive elements in an ordered group is called the positive cone, it is often denoted with G+ ; the slightly different notation G^+ is used for the positive cone together with the identity element. In this section, \le is a left-invariant order on a group G with identity element e . Beyond the home domain. No canonical parent is asserted for Linearly ordered group.

Relationships to Other Abstractions

Local relationship map for Linearly ordered groupParents 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.Linearlyordered groupDOMAINPrime abstraction: Order — presupposesOrderPRIMEPrime abstraction: Group — is a kind ofGroupPRIME

Current abstraction Linearly ordered group Domain-specific

Parents (2) — more general patterns this builds on

  • Linearly ordered group is a kind of Group Prime

    A linearly ordered group is a group with an additional translation-invariant total order.

  • Linearly ordered group presupposes Order Prime

    The identity presupposes a total order compatible with group translation.

Hierarchy paths (8) — routes to 7 parentless roots

Neighborhood in Abstraction Space

Linearly ordered group 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, Groups & Operators (42 abstractions)

Nearest neighbors

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