Generalised Free Vector Theory Overview

• GFVT is a geometric and algebraic framework that generalizes classical vector spaces by deriving scalar operations and basis properties internally.
• It establishes a unique analytic representation through a bijection between endomorphisms and coordinate tuples, enabling the construction of free objects in a categorical setting.
• The theory encompasses diverse structures—ranging from semilattices to rings—and finds applications in affine, projective, and noncommutative geometries.

Generalised Free Vector Theory (GFVT) is a geometric and algebraic framework that extends classical vector space theory by extracting and generalizing the structural principles underlying vector spaces. The paradigm replaces the standard two-sorted module-over-a-field approach with a single-sorted universal algebraic setting, emphasizing internal automorphism structures, coordinate representations, and an abstract notion of dilatations. GFVT accommodates a wide class of algebraic objects, generalizing vector spaces, and encompasses both familiar and unconventional "vector" objects, such as semilattices, schedule networks, and rings of integers, while retaining an internal structure adequate for affine and projective geometry, free-object constructions, and analytic representations (Ricci, 2019).

1. Geometric and Algebraic Foundations

Generalised Free Vector Theory is grounded in a single-component universal algebra $(A, O)$, equipped with a distinguished "basis" map $U: X \to A$ from an index set $X$. This map establishes a coordinate system analogous to classical bases, but without requiring an external scalar ring or field.

The framework is built on three axioms:

1. Analytic Representation: The map $ru: \mathrm{End}(A) \to A^X$ defined by $ru(h) = h \circ U$ is bijective. Its inverse $n = ru^{-1}: A^X \to \mathrm{End}(A)$ reinterprets tuples as endomorphisms, generalizing the construction of linear maps from column data.
2. General Dilatations: For each $d \in A$, the function $X d \circ k \in \mathrm{End}(A)$ is defined by $k(a)(x) = a$. The map $y: D \to \mathrm{End}(A)$, $y(d) = X d \circ k$, defines the set $A'$ (the endowed dilatation monoid), which generalizes scalars as a commutative submonoid of endomorphisms.
3. Dilatation Fullness: $(A, O, U)$ is dilatation full if $D = A$, so $y: A \to \mathrm{End}(A)$ is total. Classically, this recovers the ground field as the group of all scalar multiplications, but in general, $A'$ may not be a field or even a ring.

These axioms replace all explicit distributivity or field axioms, capturing scalar multiplication and its properties internally.

2. Categorical Structure and Free Objects

The collection of commutative based universal algebras (triples $(A; O; U: X \to A)$ with signature $O$ and basis $U$) forms a variety—closed under products, subalgebras, and homomorphic images—thus admitting free objects for any generating set $X$ (Ricci, 2019). The free object on $X$ is the term algebra $T(X)$, with $i: X \to T(X)$ as the basis; it universalizes the coordinate representation property and ensures every algebra in the category has a unique morphism factoring any assignment $X \to A$.

Unlike free modules, free GFVT-objects embed the scalar operations as structures derived from dilatations, not external imposition. The classical vector space appears as a special case: if $O$ consists of abelian group operations plus scalar multiplication, the resulting dilatation monoid is a field.

The relation between GFVT and "free vector lattice" constructions (as in (Jeu, 2020)) is categorical: both are realized as free objects in equational classes, characterized by universally quantified identities, and their existence is guaranteed by the general theory of varieties.

3. Analytic Representation, Dilatation Monoids, and Semi-Linear Maps

A defining property of GFVT is the analytic (matrix-like) representation: the bijection between endomorphisms and coordinate tuples. This underpins much of the geometric reasoning in the theory.

Dilatation monoids emerge as the internal scalars, acting as a commutative submonoid of $\mathrm{End}(A)$. The structure of $A'$ can be diverse (e.g., Boolean lattices, $\mathbb{N} \cup \{\infty\}$, or $\mathbb{Z}[i]$, depending on $A$).

Semi-linear transformations generalize classical semi-linearity but with respect to $A'$. A bijection $\phi: A \to A$ is semi-linear if there is an automorphism $\sigma: A' \to A'$ with $\phi \circ d \circ \phi^{-1} = \sigma(d)$ for all $d \in A'$. In relation to geometric invariance, notions preserved under all semi-linear self-maps correspond with classical affine and projective invariance when $A$ is a genuine vector space (Ricci, 2019).

4. Examples and Extensions Beyond Modules

GFVT encompasses a wide variety of algebraic structures:

Example	Carrier $A$	Endowed Dilatation Monoid $A'$
Power set semilattices	$P(X)$	Two-element Boolean lattice
CPM-PERT network semilattices	Partial schedules on $X$	$\mathbb{N} \cup \{\infty\}$ under $+$
Additive group of integers	$\mathbb{Z}$	$(\mathbb{Z}, +, \cdot)$
Gaussian integers	$\mathbb{Z}[i]$	$(\mathbb{Z}[i], +, \cdot)$

Power set semilattices interpret $A^X$ as adjacency lists for directed graphs and their endomorphisms as certain set-theoretic operations, with scalar structure emerging as the Boolean lattice. CPM-PERT networks realize GFVT objects with operations rooted in scheduling theory. Rings such as $\mathbb{Z}$ and $\mathbb{Z}[i]$ are included, demonstrating that GFVT contains, but transcends, classical module categories.

None of these examples require external module structures over a ring; instead, the existence and properties of scalars are an internal, derived phenomenon (Ricci, 2019).

5. Comparison with Free Vector Lattices and Algebras

Parallel categorical developments arise in the context of free vector lattices and free vector lattice algebras (Jeu, 2020). Here, the chain of categories (Set, VS, VL, VLA, VLA$_1$, VLA$_1^+$) supports the construction of free objects—vector spaces, vector lattices, vector lattice algebras, and their unital/positive variants—through the methodologies of universal algebra. For a given generating set $S$, the free object is the quotient of the term algebra by equational congruences, and universal properties guarantee existence and uniqueness.

This approach extends to analytic constructions, such as free Banach lattices. For a Banach space $V$, the algebraic free object is equipped with a universal seminorm and completed. Current open problems in this direction include the precise status of free $f$-algebras, where equational closure under homomorphic images is delicate and, for several variants, unresolved (Jeu, 2020).

6. Connections to Noncommutative and Free Analysis

GFVT's geometric approach to vectors generalizes naturally to noncommutative and "free" settings. Free analysis investigates analytic functions and vector fields on tuples of matrices, replacing pointwise spaces with direct sums and similarity invariance.

A recent development is the extension of potential theory to free vector fields: the "free" Poincaré lemma shows that on connected free domains, every free-curl free analytic vector field is the derivative of a potential; conversely, every analytic free map yields a free-curl free derivative (Augat, 2020). This mirrors the classical relationship between gradients and curl-free fields, but block-matrix symmetries and similarity invariance ensure that path-independence is automatic, and simply-connectedness is absorbed by the free structure.

Applications include the construction of free pluriharmonic conjugates and rational potentials, with implications for noncommutative de Rham cohomology, free-PDEs, and rational model realization.

7. Significance, Open Problems, and Outlook

Generalised Free Vector Theory unifies a broad spectrum of algebraic objects via internal coordinate and scalar structures, permitting the construction of free objects and the formulation of geometric invariants independently of external modules or fields. GFVT provides a lens through which new classes of "vector-like" objects—far more general than classical modules—are naturally accommodated, leveraging commutativity, dilatation fullness, and analytic representation as foundational principles (Ricci, 2019).

The categorical and analytic methodologies outlined in the parallel theory of free vector lattices and algebras demonstrate that rich universality properties persist beyond linear spaces, supporting Banach and lattice-theoretic generalizations (Jeu, 2020).

A major unresolved question concerns the characterization of free $f$-algebras in the language of equational classes. The existence of such free objects remains open for several variants, reflecting the limits of current universal-algebraic techniques in the context of positivity and order compatibility.

A plausible implication is that further evolution of GFVT and its categorical analogues will yield new universal constructions and invariance principles in settings that extend well beyond the familiar domain of linear algebra—potentially impacting network theory, optimization, and noncommutative geometry.