Locally Regular Algebraic Structures: A Unified Approach to Multineutral Groups and Fields

Abstract

We introduce the concept of multineutral structures, such as groups and fields, which generalize von Neumann regularity by allowing for localized units and weak inverses. We study their algebraic properties, define an associated notion of dimension, and show how every linear vector space can be extended to a multineutral field. Furthermore, we introduce an ordering on multineutral monoids and relate it to structural properties of local behavior.

Full text

Locally Regular Algebraic Structures: A
Unified Approach to Multineutral Groups and
Fields
Ondˇrej Br˚uha1
1Alpha Codes s.r.o., ondrej.bruha@alphacodes.eu
2025
Abstract
We introduce the concept of multineutral structures, such as groups
and fields, which generalize von Neumann regularity by allowing for
localized units and weak inverses. We study their algebraic properties,
define an associated notion of dimension, and show how every linear
vector space can be extended to a multineutral field. Furthermore,
we introduce an ordering on multineutral monoids and relate it to
structural properties of local behavior.
1 Introduction
Let R2be the set of real pairs. We consider the standard operations of
addition and scalar multiplication, making it a real vector space. In addition,
we define the multiplicative operation:
(a, b)·(c, d) = (ac, bd).
Under this operation, R2forms a commutative monoid, but not a group.
However, we can partition the space into equivalence classes (e.g., pairs with
one coordinate zero) that are closed under multiplication and each locally
forms a commutative group.
1

This motivates the definition of structures where group-like behavior is
preserved within equivalence classes, but not globally. Such structures exhibit
properties similar to von Neumann regularity, where for every element athere
exists a weak inverse bsuch that a=aba. In this work, we formalize this
idea through the concept of multineutral groups and extend it to multineutral
fields.
2 Multineutral Groups
Definition 2.1. Let Gbe a non-empty set equipped with a binary operation
·and an equivalence relation ∼. We say that (G, ·,∼) is a multineutral group
if the following conditions hold for all a, b, c ∈G:
(1) Commutativity: ab =ba,
(2) Associativity: a(bc)=(ab)c,
(3) For each a∈G, there exists a local unit ea∈[a] such that aea=a,
(4) For each a∈G, there exists a weak inverse a−1∈[a] such that aa−1=
ea,
(5) Class multiplication is well-defined: for all a, a′∈[a] and b, b′∈[b], it
holds that [ab] = [a′b′],
(6) There exists a global unit eG∈Gsuch that aeG=afor all a∈G,
(7) There exists an absorbing element o∈Gsuch that ao =ofor all a∈G.
Remark 2.2. We denote [a] as a class of equivalence containing element a
under relation ∼.
Theorem 2.3. [Basic Properties] Let Gbe a multineutral group. Then:
(1) The local unit eaassociated with an element a∈Gis unique.
(2) The global unit eGis unique.
(3) The weak inverse a−1of an element a∈Gis unique.
(4) The absorbing element ois unique.
2

(5) Let eaand ebbe local units for classes [a]and [b], respectively. Then
their product eaebis the local unit for the class [ab]:
eaeb=eab.
(6) There exists an element which is simultaneously a global and a local
unit.
Proof. We sketch the proof for statements (5) and (6), as the others follow
from straightforward arguments.
(5) Let a∈[a] and b∈[b] with local units eaand eb. Then:
aea=a, beb=b.
Hence,
abeaeb=ab.
Let wbe the weak inverse of ab in [ab], so multiplying both sides of
abeaeb=ab by won the right gives:
eabeaeb=eab ,
implying that eaebmust be the local unit in [ab] by uniqueness.
(6) Since eGis an element of G, it belongs to some class [g]. Then by
definition, eGsatisfies the local unit property for [g], i.e., it is a local
unit.
Definition 2.4. Let Gbe a multineutral group. A subset H⊆Gis called
amultineutral subgroup if:
(1) ab =ba for all a, b ∈H,
(2) a(bc) = (ab)cfor all a, b, c ∈H,
(3) For every equivalence class in H, there exists a local unit,
(4) Every element in Hhas a weak inverse in its class,
(5) Class multiplication is well-defined within H, i.e., for all a, a′∈[a]∩H
and b, b′∈[b]∩H, it holds that [ab] = [a′b′].
Definition 2.5. Let Gbe a multineutral group or subgroup. The number
of equivalence classes in Gis called the indoor dimension of G.
3

3 Multineutral Fields
Definition 3.1. Let Gbe a non-empty set equipped with two binary oper-
ations + and ·, and an equivalence relation ∼. We say that (G, +,·,∼) is a
multineutral field if the following axioms hold for all a, b, c ∈G:
(1) ab =ba (multiplicative commutativity),
(2) a(bc) = (ab)c(multiplicative associativity),
(3) For each a∈G, there exists a local multiplicative unit ea∈[a] such
that aea=a,
(4) For each a∈G, there exists a weak inverse a−1∈[a] such that aa−1=
ea,
(5) For all a, a′∈[a] and b, b′∈[b], we have [ab]=[a′b′],
(6) There exists a global multiplicative unit eG∈Gsuch that aeG=afor
all a∈G,
(7) a+b=b+a(additive commutativity),
(8) a+ (b+c) = (a+b) + c(additive associativity),
(9) There exists an additive unit (zero element) o∈Gsuch that a+o=a,
(10) For each a∈G, there exists −a∈Gsuch that a+ (−a) = o.
(11) For every a, b, c ∈Gis a(b+c) = ab +ac
Remark 3.2. Every multineutral field is a commutative ring with an absorb-
ing element o, which satisfies ao =ofor all a∈G. The absorbing element
coincides with the additive identity.
Definition 3.3. A subset F⊆Gis a multineutral subfield if it satisfies
the same axioms as a multineutral field with respect to the operations and
equivalence relation restricted to F.
Theorem 3.4. Every multineutral field with indoor dimension 2is a com-
mutative field. Conversely, every commutative field can be interpreted as a
multineutral field with indoor dimension 2.
4

Proof. Let Gbe a commutative field. We can define two equivalence classes:
[0] and [a] for a= 0. These are closed under multiplication, and Gsatisfies
all axioms of a multineutral field. Conversely, let Gbe a multineutral field
with indoor dimension 2. Then Ghas exactly two equivalence classes: [o]
and [eG]. The behavior within [eG] mimics that of a classical field, as all
elements there have multiplicative inverses. Therefore, Gis a commutative
field.
Remark 3.5. From the axioms, a multineutral field must have at least indoor
dimension 2.
Definition 3.6. Let B={x1, x2, . . . , xn} ⊆ Gbe a subset of a multineutral
field. We say that Bis a basis if:
(a) Pixi=eG,
(b) xixj=ofor all i=j.
The maximum number of mutually disjoint basis elements in Gis called the
outdoor dimension of G.
Theorem 3.7. Each basis element xiin a multineutral field is a local unit.
Proof. We prove the case where the outdoor dimension is 2. Let x+y=eG
and let ex,eybe the local units of xand y. Then:
x+y=eG, ex+ey=c.
Multiplying both equations gives:
exx+eyy+xey+yex=c.
Since xy =yx =o, the cross-terms vanish, and:
exx+eyy=x+y=c=eG.
Hence xand yare local units.
Definition 3.8. Let Vbe a vector space over a field F, and let {e1, . . . , ed}
be a basis of V. Then every element v∈Vcan be uniquely expressed as a
d-tuple over F.
5

Define a binary operation on Fdby:
(a1, . . . , ad)·(b1, . . . , bd)=(a1b1, . . . , adbd).
This operation is component-wise multiplication (Hadamard product). To-
gether with the usual vector addition and scalar multiplication from F, this
structure forms a multineutral field, which we call the extended vector space
over F.
Theorem 3.9. The outdoor dimension of a multineutral field constructed as
an extended vector space over Fdis equal to the vector space dimension d.
Proof. Let Gbe a multineutral field realized as Fdwith component-wise
multiplication. The canonical basis vectors ei= (0,...,1,...,0) satisfy:
d
X
i=1
ei= (1,...,1) = eG,
and
ei·ej=o= (0,...,0) for i=j.
Thus, {e1, . . . , ed}satisfies the axioms of a multineutral basis, and so the
outdoor dimension of Gis at least d.
Conversely, suppose Ghas a multineutral basis {x1, . . . , xk}with the same
properties. Since xixj=ofor i=j, each ximust lie in a distinct component
of Fd, implying k≤d.
Hence, d=kand the outdoor dimension equals the dimension of the
vector space.
4 Further Properties of Multineutral Groups
Definition 4.1. Let Gbe a multineutral group, and let [a],[b] be equivalence
classes in Gwith local units eaand eb, respectively. Define the product of
classes by:
[a][b] := [eaeb].
We call this operation class multiplication.
Remark 4.2. The product of local units is again a local unit in some equiva-
lence class (see Theorem 2.3). Therefore, the set of equivalence classes under
this operation forms a closed algebraic structure.
6

Definition 4.3. Let Mbe a non-empty set with a binary operation ·. We say
that (M, ·) is a multineutral monoid if the following holds for all a, b, c ∈M:
(1) ab =ba (commutativity),
(2) a(bc) = (ab)c(associativity),
(3) aa =a(idempotency),
(4) There exists a unit e∈Msuch that ae =a,
(5) There exists an absorbing element o∈Msuch that ao =o.
Remark 4.4. It is straightforward to show that both the unit and the ab-
sorbing element are unique in a multineutral monoid.
Theorem 4.5. Let Gbe a multineutral group. Then the set of its equivalence
classes, equipped with the class multiplication, forms a multineutral monoid.
Definition 4.6. Let Mbe a multineutral monoid, and let a, b ∈M. Define
the partial order ≥as follows:
a≥bif and only if ab =b.
Furthermore, define a∼bif ab =cfor some c /∈ {a, b}.
Theorem 4.7. Let (M, ·)be a multineutral monoid with the relation ≥as
defined above. Then (M, ≥)is a partially ordered set.
Proof. We verify the axioms of a partial order:
•Reflexivity: For all a∈M, we have aa =a, so a≥a.
•Transitivity: If a≥band b≥c, then ab =band bc =c. Then:
a(bc) = (ab)c=bc =c,
so ac =cand thus a≥c.
•Antisymmetry: If a≥band b≥a, then ab =band ba =a. By
commutativity, ab =ba, so a=b.
7

Theorem 4.8. In any multineutral monoid:
•The unit element eis the maximum element with respect to ≥.
•The absorbing element ois the minimum element with respect to ≥.
Proof. By definition, for any a∈M:
ae =a⇒a≤e,
and
ao =o⇒a≥o.
5 Conclusion
In this work, we introduced the concept of multineutral structures, which
generalize classical algebraic systems such as groups, rings, and fields by
incorporating localized neutral elements and weak inverses. Inspired by von
Neumann regularity, our structures allow for the decomposition of a global
algebraic system into locally invertible components that respect a common
equivalence structure.
We have shown:
•How multineutral groups and fields can be defined using local and
global units,
•That every field can be interpreted as a multineutral field with indoor
dimension 2,
•That extended vector spaces with component-wise multiplication are
naturally multineutral fields,
•That class structures derived from equivalence classes form multineu-
tral monoids,
•That such monoids can be endowed with a natural partial order reflect-
ing absorption and dominance between classes.
8

Multineutral structures provide a framework for studying local proper-
ties of algebraic systems where global inverses or global identities may not
exist. This has potential applications in areas involving partial invertibility,
structured decompositions, or nonstandard field extensions.
Future work may include:
•Categorical formulations of multineutral structures,
•Study of homomorphisms and morphisms between such structures,
•Applications in algebraic computation and symbolic systems where par-
tial operations are natural.
References
[1] J. von Neumann, Continuous Geometry, Princeton University Press,
1960.
[2] T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag,
2001.
[3] S. Lang, Algebra, Springer New York, NY, 2005
[4] S. Lang, Undergraduate Algebra, Springer New York, NY, 2010
[5] S. Mac Lane, Categories for the Working Mathematician, Springer New
York, NY, 2010
9