Linear Equivalent (LE)

• Linear Equivalent (LE) is a formal equivalence framework that preserves key algebraic, logical, and operational structures under linear or semi-linear transformations.
• It underpins applications in coding theory and lattice expansion logics by enabling the classification of linear codes and establishing canonical correspondences in logical systems.
• In engineering, LE informs system identification, adaptive control, and bifurcation analysis in power systems and linear complementarity problems using robust parametrization techniques.

The term "Linear Equivalent" (LE) is systematically deployed across several research domains, denoting a formal equivalence relation or parametric representation that preserves specific algebraic, logical, or physical structures under linear or semi-linear transformations. Such equivalence relations are foundational both in theoretical mathematics (coding theory, algebra, logic) and in engineering applications (power systems, complementarity problems). The following sections detail the principal usages and technical frameworks for LE, synthesizing contemporary research.

1. Linear Equivalence in Coding Theory

In the theory of error-correcting codes over finite fields, LE describes a canonical equivalence between linear codes via the action of the monomial group. Given a finite field $\F_q$, the space $\F_q^n$ is the $n$-dimensional coordinate space. An $[n,k]$-linear code $C$ is a $k$-dimensional subspace of $\F_q^n$. Two codes $C$, $C'$ are linearly equivalent if there exists a monomial matrix $M \in \Mon(n,q)$ such that $C' = CM = \{cM : c \in C\}$, where $\Mon(n,q)$ consists of all $n\times n$ matrices with one nonzero field element per row and column; that is, the composition of a permutation and a scaling by field units.

The concept extends to semi-linear equivalence: $C$ and $C'$ are semi-linearly equivalent if there exists $M\in\Mon(n,q)$ and a field automorphism $\sigma\in\Aut(\F_q)$ such that $C' = \{ \sigma(cM) : c\in C \}$. Ball and Dixon's theorem establishes that linear equivalence implies semi-linear equivalence, but not vice versa; thus, LE forms robust equivalence classes central to the classification and enumeration of linear codes (Ball et al., 2021).

2. LE in Lattice Expansion Logics

The notion of LE is foundational in the algebraic and correspondence-theoretic paper of normal lattice expansions (LE-algebras) and associated logical systems. Here, LE refers to both the syntactic language $L_\ell_e(F,G)$, built from monotone/antitone connectives (with prescribed arities and order-types), and the algebraic structures (bounded lattices with such operations).

In correspondence theory, LE-logics are analyzed via canonical extensions (perfect LE-algebras), which are complete lattices generated by join- and meet-irreducible elements. The LE property ensures the preservation of normality conditions across extensions, making LE-algebras a universal semantic base for modal, substructural, intuitionistic, and distributive logics (Palmigiano et al., 2 May 2024).

The inverse correspondence problem for LE-logics—computing the unique LE-inequality from first-order frame conditions within a decidable syntactic fragment—is solved via refined ALBA algorithms. This recursive connection between syntactic (algebraic) and semantic (frame-theoretic) conditions exemplifies the reach of LE beyond classical modal logic.

3. LE in Linear Complementarity Problems

Within the context of linear complementarity problems (LCPs), LE formalizes an equivalence relation—Linear–Equivalence—on LCPs characterized by matrix $M$ and vector $q$. Two LCPs $(M,q)$ and $(N,r)$ are LE if there exist homeomorphisms $\varphi$, $\psi$ of $\R^n$ such that $\psi$ permutes the $2^n$ orthants (inducing a Boolean automorphism), and the piecewise-linear maps $f_M$, $f_N$ satisfy $f_M = \varphi \circ f_N \circ \psi$.

Algebraic criteria for LE involve combinatorial properties of families of complementary cones associated with each problem. In particular, in the planar case ($n=2$), pairs of LCPs are LE iff their matrices share principal minors with nonzero matching signs—yielding a Boolean-algebraic classification that directly informs the bifurcation structure of solutions, and enabling the translation of smooth bifurcation theory to the non-smooth regime (Castaños et al., 2019).

4. LE in the Variational Bicomplex and Differential Geometry

In the calculus of variations, LE commonly refers to Lepage Equivalents—differential forms encoding the extremal structure of Lagrangians in jet bundles over manifolds. The Lepage equivalent satisfies the closure property: $d\Theta_\lambda=0$ if and only if the Euler–Lagrange form vanishes, i.e., the Lagrangian is null (Saunders, 2023).

The construction necessitates homotopy operators for the horizontal differential in the variational bicomplex. Local versions use vertical endomorphisms, while global Lepage equivalents require a symmetric linear connection to assemble homotopy operators and vertical tensors. The canonical formulas express the LE as an explicit sum involving derivatives and contact forms, recovering the classical Poincaré–Cartan form in the first-order case.

5. LE in Power Systems: Thevenin Linear Equivalent

In electrical power engineering, the LE often designates the linear Thevenin equivalent—an abstract source and impedance parameterization that models the network as seen from a point of common coupling (PCC). The LE parameters $(V_{\rm th}, R_{\rm th}, X_{\rm th})$ are estimated online using recursive least squares (RLS), both in nonlinear (Gauss–Newton iteration) and linearized forms. Measurement snapshots of voltage and current phasors at the PCC are sufficient for rapid convergence; the algorithms exhibit sub-percent estimation error with two to five samples.

The estimation of the LE informs adaptive control, impedance matching, fault detection, and voltage-stability margin monitoring. The underlying algorithms rely on the solution of parameterized linear equations, and their efficacy depends on the diversity of operating points sampled. This suggests the generality and operational significance of LE in modern power systems featuring distributed generation and reconfigurable networks (Hashmi et al., 2016).

6. Applications and Theoretical Implications

LE structures unify a broad spectrum of technical domains:

• In coding theory, LE provides the foundation for code equivalence classification, automorphism group computation, and MDS code enumeration.
• In lattice expansions and logic, LE enables the uniform algebraic analysis and correspondence between syntactic rules and semantic frame conditions.
• In complementarity problems, LE enables the stratification of solution sets and bifurcation types, allowing for the design and prediction of non-smooth dynamical transitions.
• In variational calculus, LE secures the existence of closed extremal forms and thereby underpins a major aspect of the geometric theory of Lagrangian systems.
• In power engineering, LE (as Thevenin equivalent) is pivotal for real-time system identification, adaptive control, and grid stability.

A plausible implication is that LE formalism serves as a categorical bridge across linear structures in mathematics and engineering, offering both classification invariants and constructive parameterizations.

7. Summary Table: LE Usage Across Domains

Domain	Definition/Transformation	Key Outcome
Coding Theory	Monomial/semi-linear map on codes	Equivalence classification
Lattice Expansion Logics	Normal lattice expansions	Correspondence theory
Complementarity Problems	Cone/orthant-preserving maps	Bifurcation stratification
Variational Geometry	Lepage equivalent differential	Closure property on forms
Power Systems	Linear Thevenin parameterization	System identification

The LE construct, though instantiated differently according to context, consistently encodes a canonical linear relationship or equivalence—underpinning operational, algebraic, and logical analysis at both abstract and applied levels.