Traction Theory introduces a novel algebraic structure called a traction, designed to be total, reversible, and emergent. Unlike classical algebras, tractions treat operations as primary elements and avoid exceptions by allowing unevaluated symbolic expressions as valid members. The system incorporates the additive identity 0 and its multiplicative reciprocal ω explicitly, ensuring consistency and maximal tractability without structural collapse. This paper outlines the motivation, core definitions, and operations of addition, multiplication, and exponentiation in tractions, emphasizing preservation of invariants and information.
Algebra fundamentally involves the symbolic manipulation of expressions under declared invariants. Valid manipulations preserve these invariants while uncovering underlying structure. define an algebra as maximally tractable if it admits every transformation proven to preserve all declared invariants of an expression. When projection is defined (e.g., to numerical values), it must map invariant-preserving equivalent expressions to equal projections.
An algebra is consistent if it prohibits manipulations that would invalidate its invariants. A traction is an algebra engineered to be both consistent and maximally tractable: it allows all invariant-preserving transformations and forbids those that violate invariants.
Classical algebras often suffer from exceptions, such as division by zero or undefined operations, leading to loss of tractability. Tractions address this by creating a "wheel-like" structure where the additive identity 0 and its multiplicative reciprocal ω are valid elements, ensuring an exception-free system.
Traction Theory implements Constructive Operational Type Theory (COTT), where operations are foundational, and evaluation is not mandatory. For details on COTT, see sibarum.github.io/cott. Tractions are invariant-agnostic at the core, with universal invariants limited to operational identities; other invariants are contextual and arise in projections. Equality is structural rather than purely equational, and terminal constructions maximize information compression without erasure.
This contrasts with related structures like wheels, which introduce a nullity element ⊥ leading to collapse, or meadows, which handle undefinedness differently. Tractions aim for totality, respecting universal invariants while preserving symbolic information.
A traction is a totalized algebra with two group-like structures (for addition and multiplication), a distinguished inverse pair (0, ω), and boundary-sensitive reassociation rules. Types in tractions are extended operationally: they include numeric members and are closed under all admissible operations. Symbolic expressions are canonical representatives of distinct elements, eliminating undefinedness—expressions simply may not project to single numeric values.
Traction algebra is not an extension of ℝ or ℂ zero is non-absorbing and information-preserving, necessitating adjustments to the type theory. Operations are total and reversible, with recovery laws ensuring invertibility.
Key principles:
Addition in a traction forms a closed Abelian group.
For every term t, there exists a corresponding additive inverse, denoted –t, satisfying the involution:
Negation is structural: –t constructs a new symbolic term without evaluation or simplification.
Addition is a total binary operation on terms. For all terms a,b,c:
If a+b=c, then the recovery laws hold:
Addition further satisfies associativity and commutativity:
There exists a term 0 such that for every term t:
From the recovery laws:
so 0 is the additive identity.
Multiplication forms a near-Abelian group, with subtle deviations due to constraints on identities.
For every term t, there exists a corresponding multiplicative inverse, denoted t^–1, satisfying the involution:
Inversion is structural, constructing a new term without evaluation.
Multiplication is a total binary operation on terms. For all terms a,b,c:
If a·b=c, then the recovery laws hold:
Multiplication further satisfies associativity and commutativity:
There exists a term 1 such that for every term t,
From the recovery laws:
The inverse of 1 is itself:
Zero is non-absorbing and preserves information in multiplications. It has a multiplicative inverse, aliased as ω:
They satisfy:
The additive identity is unique.
For any unassigned variable x:
Exponentiation is a total binary operation on terms, written a^b . It extends classical exponentiation while ensuring totality even when the base is 0 or the exponent involves ω, without absorption or collapse.
For all terms a,b,c, where the expressions are defined in the usual sense (non-zero base or non-boundary cases):
(where / is multiplication by the multiplicative inverse)
To establish the zero lift in a way that preserves reversibility and totality, we define exponentiation and its structural logarithm base 0 on key boundary cases as follows. These rules extend the familiar exponentiation laws while ensuring that log_0 serves as a recovery operation (right-inverse) for terms of the form 0^x .
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General Rule
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Zero-Lifted Expression
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Structural Log Inverse
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b^0=1
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0^0=1
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log_{0}1=0
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b^1=b
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0^1=0
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log_{0}0=1
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b^–1=1/b
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0^–1=ω
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log_{0}ω=–1
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The first two rows maintain continuity with classical conventions. The third row lifts the inverse of zero to ω, consistent with the multiplicative structure 0·ω=1 and the classic negative exponent rule.
The same conventions must be followed for ω, ensuring symmetric treatment of the distinguished inverse pair (0, ω). These rules extend the familiar exponentiation laws while guaranteeing that log_ω serves as a recovery operation (right-inverse, and two-sided in the final case) for terms of the form ω^x.
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General Rule
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ω-Lifted Expression
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Structural Log Inverse
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b^0=1
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ω^0=1
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log_{ω}1=0
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b^1=b
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ω^1=ω
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log_{ω}ω=1
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b^–1=1/b
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ω^–1=0
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log_{ω}0=–1
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The following identities must hold structurally for all terms in any traction implementation. They extend the classical exponentiation and logarithm properties to the totalized setting, ensuring that the structural logarithms base 0 and base ω behave as proper inverses without exception. These are consequences of the operational definitions and recovery laws already introduced; they are reiterated here for clarity and to affirm that standard algebraic manipulations remain valid away from boundaries.
For base v∈{0,ω} and all terms a,b,c:
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v^(log_{v}a)=a
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log_{v}(v^a)=a
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v^a·v^b=v^(a+b)
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log_{v}(a·b)
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v^a/v^b
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log_{v}(a/b)
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(v^a)^b
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b·log_{v}a
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v^a
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log_{v}a
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Therefore: