Hemiring-Valued Pseudonormed Rings

• Hemiring-valued pseudonormed rings are rings endowed with a size function taking values in an ordered hemiring, ensuring nondegeneracy, subadditivity, and submultiplicativity.
• They extend classical metrics by inducing hemiring-valued distances and generalizing convergence tests like the Cauchy Condensation and ratio tests in analytic settings.
• The framework generalizes Albert’s theorem by showing that every finite-dimensional algebra can be pseudonormed, while connecting to valuation theory and pseudo-absolute value concepts.

Searching arXiv for the cited paper and closely related work to ground the article. {"^^^^^^^^2query2 OR \2^^^^^^^^"Hemiring-valued pseudonormed rings\""} Hemiring-valued pseudonormed rings are rings equipped with a size function taking values not in PRESERVED_PLACEHOLDER_^^^^2query2^^^^ but in an ordered hemiring, so that nonnegativity, definiteness, subadditivity, and submultiplicativity are formulated internally to the hemiring order. In the form developed in “Hemiring-valued pseudonormed rings” (&&&^^^^2query2^^^^&&&), the notion extends the framework of “Magma-valued metric spaces” (Nasehpour, 2022), generalizes Albert’s theorem on normability of finite-dimensional algebras, and connects normed-ring methods with valuation-theoretic examples. The paper also develops analytic consequences in ordered rings and fields, including a Cauchy Condensation Test for Cauchy complete fields and a ratio test for ring-valued normed groups (&&&^^^^2query2^^^^&&&).

^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^. Ordered hemirings and the definition of pseudonorm

An ordered hemiring is a structure

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^

such that $(H,+,0,\le)$ is a commutative ordered monoid, $(H,\cdot)$ is a semigroup, multiplication is monotone with respect to nonnegative elements on both sides, multiplication distributes over addition from both sides, and $0$ is absorbing: $0h=h0=0$ for all $h\in H$ (&&&^^^^2query2^^^^&&&). A hemiring is commutative if its multiplication is commutative. The paper also uses the notion of a strictly ordered hemiring $(H,<)$, meaning an ordered hemiring together with a strict order compatible with addition and with left and right multiplication by positive elements. If $\le$ is total and $(H,<)$ is an ordered hemiring, then PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2query2^^^^ is entire:

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2id:(Nasehpour, 2 Aug 2025) OR \2^^^^

This provides the ordered algebraic target in which all norm inequalities are interpreted (&&&^^^^2query2^^^^&&&).

Given a ring PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \22^^^^ and an ordered hemiring PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^3, an PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^4-pseudonormed ring is a ring PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^5 together with a function

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^6

such that, for all PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^7,

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^8

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^9

$(H,+,0,\le)$^^^^2query2^^^^

If equality holds multiplicatively for all $(H,+,0,\le)$^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^, so that $(H,+,0,\le)$2, then $(H,+,0,\le)$3 is called an $(H,+,0,\le)$4-normed ring and $(H,+,0,\le)$5 its $(H,+,0,\le)$6-norm (&&&^^^^2query2^^^^&&&). Despite the term “pseudonorm,” the paper imposes the nondegeneracy axiom $(H,+,0,\le)$7, which is typical of a norm; the term “pseudo” reflects that the codomain is a general hemiring rather than necessarily $(H,+,0,\le)$8 (&&&^^^^2query2^^^^&&&).

No additional homogeneity axiom is imposed when $(H,+,0,\le)$9 is an algebra over a base field or semiring. In the finite-dimensional existence theorem, the pseudonorm on the algebra is instead built by summing the pseudonorms of coordinates in a chosen basis (&&&^^^^2query2^^^^&&&).

2. Metric interpretation and relation to classical norms and valuations

An $(H,\cdot)$^^^^2query2^^^^-pseudonorm induces an $(H,\cdot)$^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^-valued distance by

$(H,\cdot)$2

and subadditivity yields the triangle inequality

$(H,\cdot)$3

This places hemiring-valued pseudonormed rings inside the broader framework of magma-valued metric spaces introduced in (Nasehpour, 2022), where convergence, Cauchy sequences, and completeness are studied for metrics valued in ordered unital magmas. In that earlier framework, density of the value object is the key hypothesis ensuring uniqueness of limits, the implication “convergent $(H,\cdot)$4 Cauchy,” and the subsequence criterion for Cauchy convergence (Nasehpour, 2022).

Classical normed structures appear as special cases. If $(H,\cdot)$5 is an ordered ring and one sets

$(H,\cdot)$6

then $(H,\cdot)$7 gives an $(H,\cdot)$8-metric space in the generalized sense used in the paper (&&&^^^^2query2^^^^&&&). With $(H,\cdot)$9 and the usual order, an $0$^^^^2query2^^^^-pseudonorm is just an ordinary nonnegative real-valued norm satisfying the triangle inequality and a submultiplicative inequality (&&&^^^^2query2^^^^&&&).

Valuation-theoretic examples are equally central. Let $0$^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^ be a totally ordered group, and let $0$2 with $0$3, $0$4 for all $0$5, addition defined by $0$6, and multiplication given by the group law extended by $0$7. Then $0$8 is a division semiring, and a valuation

$0$9

satisfies

$0h=h0=0$^^^^2query2^^^^

Thus valuation rings and non-Archimedean normed structures fit exactly into the hemiring-valued norm framework (&&&^^^^2query2^^^^&&&).

A related but distinct line of work appears in “Pseudo-absolute values: foundations” (&&&^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^4&&&). There, a pseudo-absolute value on a field is a map to $0h=h0=0$^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^ satisfying subadditivity and a “safe” multiplicativity condition outside the exceptional pair $0h=h0=0$2. The paper explicitly observes that this framework naturally adapts to a hemiring-valued pseudonormed ring viewpoint, with finiteness subring and kernel ideal defined from the values of the pseudonorm (&&&^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^4&&&).

3. Finite-dimensional algebras and the generalization of Albert’s theorem

A central theorem states that if $0h=h0=0$3 is a total ordering on $0h=h0=0$4, $0h=h0=0$5 is a totally ordered commutative hemiring, and $0h=h0=0$6 is a field equipped with an $0h=h0=0$7-pseudonorm $0h=h0=0$8, then every finite-dimensional $0h=h0=0$9-algebra $h\in H$^^^^2query2^^^^ admits the structure of an $h\in H$^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^-pseudonormed ring (&&&^^^^2query2^^^^&&&). This is the paper’s generalization of Albert’s classical result that every finite-dimensional algebra can be normed.

The construction is explicit. If $h\in H$2 is an $h\in H$3-basis of $h\in H$4 and

$h\in H$5

the pseudonorm is defined by

$h\in H$6

Nonnegativity, definiteness, and subadditivity are immediate from the corresponding properties on $h\in H$7 (&&&^^^^2query2^^^^&&&). To control multiplication, one writes

$h\in H$8

and sets

$h\in H$9

Then for

$(H,<)$^^^^2query2^^^^

one obtains

$(H,<)$^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^

After rescaling by the factor $(H,<)$2, namely

$(H,<)$3

one has

$(H,<)$4

so $(H,<)$5 becomes an $(H,<)$6-pseudonormed ring (&&&^^^^2query2^^^^&&&).

The role of the hemiring hypotheses is explicit in the construction. Commutativity of $(H,<)$7 is used to control the product $(H,<)$8 and to scale uniformly, while total order is used to make maxima and comparison estimates meaningful. No completeness assumption is needed (&&&^^^^2query2^^^^&&&). When $(H,<)$9 with the usual order and $\le$^^^^2query2^^^^ is the ordinary absolute value, the argument recovers the standard proof that every finite-dimensional algebra over a normed field is pseudonormable, extending Albert’s theorem for real algebras to the hemiring-valued setting (&&&^^^^2query2^^^^&&&).

4. Dense and shrinkable hemirings

The analysis developed around hemiring-valued pseudonorms depends on additional order-theoretic hypotheses on the value object. An ordered hemiring $\le$^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^ is called shrinkable if, for every positive $\le$2, there exist positive elements $\le$3 such that

$\le$4

An ordered hemiring $\le$5 is called dense if for every $\le$6 there exist $\le$7 with

$\le$8

Intuitively, density is an $\le$9-splitting property, while shrinkability allows positive elements to be made small on either side under multiplication (&&&^^^^2query2^^^^&&&).

A basic structural result is that every dense ordered division semiring is shrinkable. If $(H,<)$^^^^2query2^^^^ is an ordered division semiring, $(H,<)$^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^, and $(H,<)$2, density provides $(H,<)$3 with $(H,<)$4, and one sets

$(H,<)$5

Then

$(H,<)$6

which is precisely shrinkability (&&&^^^^2query2^^^^&&&).

The paper records several examples. If $(H,<)$7 is a totally ordered group and $(H,<)$8 is equipped with addition by $(H,<)$9 and multiplication by the group operation extended with PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2query2query2^^^^ absorbing, then PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2query2id:(Nasehpour, 2 Aug 2025) OR \2^^^^ is a division semiring and is shrinkable. DeMarr division rings are shrinkable. If PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2query22^^^^ is an ordered field, then its nonnegative cone PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2query2^^^^3 is a totally ordered semifield, and for any PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2query2^^^^4 one can take PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2query2^^^^5 to see density, hence shrinkability. For a prime PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2query2^^^^6, the ring PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2query2^^^^7 is dense and shrinkable (&&&^^^^2query2^^^^&&&).

By contrast, the natural-number hemiring PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2query2^^^^8 with its usual order is not dense, since there are no positive PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2query2^^^^9 with PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2id:(Nasehpour, 2 Aug 2025) OR \2query2^^^^, and it is not a division semiring. Consequently it is not shrinkable under the paper’s definition (&&&^^^^2query2^^^^&&&). This addresses a common misconception: the theory is not designed for arbitrary ordered hemirings, and the main limit and series results do not hold uniformly without density and, in several places, shrinkability.

5. Completeness, condensation, and ratio tests

The analytic part of the theory is developed in ordered rings and ring-valued normed groups. For an ordered ring PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2id:(Nasehpour, 2 Aug 2025) OR \2id:(Nasehpour, 2 Aug 2025) OR \2^^^^ with total order, a sequence PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2id:(Nasehpour, 2 Aug 2025) OR \22^^^^ is Cauchy if for every PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2id:(Nasehpour, 2 Aug 2025) OR \2^^^^3 there exists PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2id:(Nasehpour, 2 Aug 2025) OR \2^^^^4 such that for all PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2id:(Nasehpour, 2 Aug 2025) OR \2^^^^5,

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2id:(Nasehpour, 2 Aug 2025) OR \2^^^^6

where

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2id:(Nasehpour, 2 Aug 2025) OR \2^^^^7

The ring is Cauchy complete if every Cauchy sequence converges in PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2id:(Nasehpour, 2 Aug 2025) OR \2^^^^8 (&&&^^^^2query2^^^^&&&).

Under density and Cauchy completeness, the paper proves a full Cauchy Condensation Test. If PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2id:(Nasehpour, 2 Aug 2025) OR \2^^^^9 is a dense and Cauchy complete ordered ring with PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \22query2^^^^ and PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \22id:(Nasehpour, 2 Aug 2025) OR \2^^^^ is positive and decreasing, then

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \222^^^^

The same statement holds for totally ordered, Cauchy complete fields (&&&^^^^2query2^^^^&&&). The proof follows the classical argument but uses density to split PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \22^^^^3 as PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \22^^^^4 and completeness to pass from Cauchy control to convergence. The paper also proves a squeeze test for series in the same setting: if PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \22^^^^5 eventually and both PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \22^^^^6 and PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \22^^^^7 converge, then PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \22^^^^8 converges (&&&^^^^2query2^^^^&&&).

In the fourth section, the paper generalizes Bernoulli’s inequality to ordered semirings. For elements PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \22^^^^9 with PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^3^^^^2query2^^^^ and PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^3^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^,

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^32

and in particular, for PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^33,

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^34

This estimate is then used in a ratio test for ring-valued normed groups (&&&^^^^2query2^^^^&&&).

The ratio test is stated as follows. Let PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^35 be an ordered commutative ring with PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^36 that is dense and shrinkable. Suppose PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^37 satisfies that PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^38 is invertible and PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^39 in PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^4^^^^2query2^^^^. Let PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^4^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^ be a Cauchy complete PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^42-normed abelian group and let PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^43 satisfy

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^44

Then the series

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^45

converges in PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^46 (&&&^^^^2query2^^^^&&&). The paper notes both the classical case, with PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^47 and PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^48 a Banach space, and a non-Archimedean ordered-field case with PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^49 as illustrative examples (&&&^^^^2query2^^^^&&&).

6. Context, scope, and limitations

Hemiring-valued pseudonormed rings were already introduced in “Magma-valued metric spaces” (Nasehpour, 2022), where they were presented as a generalization of pseudonormed rings and valuation domains. That paper also established the surrounding convergence theory for magma-valued metrics, including the density-based implications needed later for normed groups, rings, and series. The 2^^^^2query22^^^^5 paper develops this line further by organizing the algebraic theory around ordered hemirings, proving the finite-dimensional algebra theorem, and adding shrinkability, condensation, and ratio-test results (&&&^^^^2query2^^^^&&&).

The framework sits naturally beside the theory of pseudo-absolute values on fields. In (&&&^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^4&&&), one views PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^5^^^^2query2^^^^ as a commutative hemiring with an additional top element PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^5^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^, and a pseudo-absolute value is a map

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^52

satisfying

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^53

subadditivity, and multiplicativity outside the excluded pair PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^54. For such a map one defines

PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^55

and in the field case PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^56 is a valuation ring, PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^57 is its maximal ideal, and PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^58 induces an ordinary absolute value on the residue field (&&&^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^4&&&). The paper explicitly identifies this as a hemiring viewpoint.

Several limitations are explicit in the literature. The key analysis-type results for hemiring-valued normed rings and groups require density in order to split PRESERVED_PLACEHOLDER_^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^59 and derive Cauchy control, and require shrinkability in several multiplicative limit arguments. For the multiplication-of-limits theorem on sequences, the value ring is also assumed to be a join-semilattice. Completeness is needed whenever Cauchy estimates are converted into actual convergence (&&&^^^^2query2^^^^&&&). Likewise, the pseudo-absolute-value framework develops local and global analytic spaces primarily at the level of compact Hausdorff topological spaces; a full analytic structure sheaf on the general spaces is not attached beyond the model approach, and extending the field-based results to general rings requires additional algebraic hypotheses (&&&^^^^2id:(Nasehpour, 2 Aug 2025) OR \2^^^^4&&&).

Within these constraints, hemiring-valued pseudonormed rings provide a single formalism encompassing ordinary norms, valuation norms with ultrametric behavior, and value structures such as totally ordered semifields and idempotent-style semirings arising from ordered groups. The resulting theory is algebraic in definition but analytic in consequence: it supports metric notions, convergence and completeness, extension of pseudonorms to finite-dimensional algebras, and classical tests for infinite series in settings that go beyond the real numbers (&&&^^^^2query2^^^^&&&).