Scalable monoids and quantity calculus

We define scalable monoids and prove their fundamental properties. Congruence relations on scalable monoids, direct and tensor products, subalgebras and homomorphic images of scalable monoids, and unit elements of scalable monoids are defined and investigated. A quantity space is defined as a commutative scalable monoid over a field, admitting a finite basis similar to a basis for a free abelian group. Observations relating to the theory of measurement of physical quantities accompany the results about scalable monoids. We conclude that the algebraic theory of scalable monoids and quantity spaces provides a rigorous foundation for quantity calculus.


Introduction and historical background
Equations such as E = mv 2 2 or ∂ T ∂t = κ ∂ 2 T ∂ x 2 , used to express physical laws, describe relationships between scalars, commonly real numbers. An alternative interpretation is possible, however. Since the scalars assigned to the variables in these equations are numerical measures of certain quantities, the equations express relationships between these quantities as well. For example, E = mv 2 2 can also be interpreted as describing a relation between an energy E, a mass m and a velocity v-three underlying physical quantities, whose existence and properties do not depend on scalars used to represent them. With this interpretation, though, mv 2 2 and similar expressions will have meaning only if operations on quantities, corresponding to operations on numbers, are defined. In other words, an appropriate way of calculating with quantities, a quantity calculus, needs to be available. denotes the product of a number and a magnitude. As shown in Sect. 2.6, these identities are fundamental in modern quantity calculus as well.
If p and q are magnitudes of the same kind, and there is some magnitude u of this kind and some numbers m, n such that p = m ×u and q = n ×u, then p and q are said to be "commensurable". The ratio of magnitudes p : q can then be represented by the ratio of numbers m : n, assumed to be unique (unlike the two numbers specifying the ratio). However, magnitudes may also be "relatively prime"; it may happen that p : q cannot be expressed as m : n for any numbers m, n because there are no m, n, u such that p = m × u and q = n × u. In view of the Pythagorean philosophical conviction of the primacy of numbers, the discovery of examples of such "incommensurable" magnitudes created a deep crisis in early Greek mathematics [8], a crisis that also affected the foundations of geometry.
If ratios of arithmoi do not always suffice to represent ratios of magnitudes, it seems that it would not always be possible to express in terms of arithmoi the fact that two ratios of magnitudes are equal, as are the ratios of the lengths of corresponding sides of similar triangles. This difficulty was resolved by Eudoxus, who realized that a "proportion", that is, a relation among magnitudes of the form " p is to q as p is to q ", conveniently denoted p : q :: p : q , can be defined numerically even if there is no pair of ratios of arithmoi m : n and m : n corresponding to p : q and p : q , respectively, so that p : q :: p : q cannot be inferred from m : n = m : n . Specifically, as described in Book V of the Elements, Eudoxus invented an ingenious indirect way of determining if p : q :: p : q in terms of nothing but arithmoi by means of a construction similar to the Dedekind cut [8]. Using modern terminology, one can say that Eudoxus defined an equivalence relation :: between pairs of magnitudes of the same kind in terms of positive integers, and as a consequence it became possible to conceptualize in terms of arithmoi not only ratios of magnitudes corresponding to rational numbers but also ratios of magnitudes corresponding to irrational numbers. Eudoxus thus reconciled the continuum of magnitudes with the discrete arithmoi, but in retrospect this feat reduced the incentive to rethink the Greek notion of number, to generalize the arithmoi.
To summarize, Greek mathematicians used two notions of muchness, and built a theoretical system around each notion. These systems were connected by relationships of the form q = n × u, where q is a magnitude, n a number and u a magnitude of the same kind as q, foreboding from the distant past Maxwell's quantity formula q = {q} [q], although Euclid wisely did not propose to define magnitudes in terms of units and numbers.
The modern theory of numbers dramatically extends the theory of numbers in the Elements. Many types of numbers other than positive integers have been added, and the notion of a number as an element of an algebraic system has come to the forefront. The modern notion of number was not developed by a straight-forward extension of the concept of arithmos, however; the initial development of the new notion of number during the Renaissance was strongly inspired by the ancient theory of magnitudes.
The beginning of the Renaissance saw renewed interest in the classical Greek theories of magnitudes and numbers as known from Euclid's Elements, but later these two notions gradually fused into that of a real number. Malet [9] remarks: As far as we know, not only was the neat and consistent separation between the Euclidean notions of numbers and magnitudes preserved in Latin medieval translations [...], but these notions were still regularly taught in the major schools of Western Europe in the second half of the 15th century. By the second half of the 17th century, however, the distinction between the classical notions of (natural) numbers and continuous geometrical magnitudes was largely gone, as were the notions themselves. [pp. [64][65] The force driving this transformation was the need for a continuum of numbers as a basis for computation; the discrete arithmoi were not sufficient. As magnitudes of the same kind form a continuum, the idea emerged that numbers should be regarded as an aspect of magnitudes. "Number is to magnitude as wetness is to water" said Stevin in L'Arithmétique [10], published 1585, and defined a number as "that by which one can tell the quantity of anything" (cela, par lequel s'explique la quantité de chascune chose) [Definition II]. Thus, numbers were seen to form a continuum by virtue of their intimate association with magnitudes.
Stevin's definition of a number is rather vague, and it is difficult to see how a magnitude can be associated with a definite number, considering that the numerical measure of a magnitude depends on a choice of a unit magnitude. The notion of number was, however, refined during the 17th century. In La Geometrie [11], where Descartes laid the groundwork for analytic geometry, he implicitly identified numbers with ratios of two magnitudes, namely lengths of line segments, one of which was considered to have unit length, and in Universal Arithmetick [12] Newton, who had studied both Euclid and Descartes, defined a number as follows: By Number we mean, not so much a Multitude of Unities, as the abstracted Ratio of any Quantity, to another Quantity of the same Kind, which we take for Unity. [p. 2] By assigning the number 1 to a unit quantity, the representation of quantities by numbers is normalized, addressing a problem with Stevin's definition. Also, a ratio of quantities of the same kind is a "dimensionless" quantity. Systems of such quantities contain a canonical unit quantity 1, and addition, subtraction, multiplication and division of dimensionless quantities yield dimensionless quantities. Hence, a number and the corresponding dimensionless quantity are quite similar, though Newton hints at a difference by calling numbers "abstracted" ratios of quantities.
Magnitudes, or "dimensionful" quantities, were thus needed only as a scaffolding for the new notion of numbers, and when this notion had been established its origins fell into oblivion and magnitudes fell out of fashion. The tradition from Euclid paled away, but the idea that numbers specify quantities relative to other quantities remained, as in [13]. A new theory of quantities originated from this idea.
While the Greek theory of magnitudes derived from geometry, the new theory of quantities found applications in mathematical physics, a branch of science that emerged in the 18th century. In The Analytic Theory of Heat [6], published in 1822 as Théorie Analytique de la Chaleur, Fourier explains how physical quantities relate to the numbers in his equations: In order to measure these quantities and express them numerically, they must be compared with different kinds of units, five in number, namely, the unit of length, the unit of time, that of temperature, that of weight, and finally the unit which serves to measure quantities of heat. [pp. 126-127] We recognize here the ideas that there are quantities of different kinds and that the number associated with a quantity depends on the choice of a unit quantity of the same kind.
Using the modern notion of, for example, a real number, we can generalize relationships of the form q = n × u, where n is an arithmos and u is a magnitude that measures (divides) q, to relationships of the form q = μ · u, where u is a freely chosen unit quantity of the same kind as q, and μ is the measure of q relative to u, a number specifying the size of q compared to u. If q = μ · u then μ is determined by q and u, and we may write μ = f (q, u) as μ = q/u.
Fourier realized that the measure of a quantity may be defined in terms of measures of other quantities, in turn dependent on the units for these quantities. For example, the measure of a velocity depends on a unit of length u and a unit of time u t since a velocity is defined in terms of a length and a time, and the measure of an area indirectly depends on a unit of length u .
Formally, let the measure μ v of a velocity v relative to u v be given by where F (x, y) = x y −1 , and let the measure μ a of the area a of a rectangle relative to u a be given by Generalizing the magnitude identity m × (n × u) = mn × u, we have M · (N · u) = M N · u for any real numbers M, N . Thus, if q = μ · u and M > 0 then q = MμM −1 · u = Mμ · M −1 · u , so Mμ = q/ M −1 · u , so it follows from the definitions of F and G that, for any non-zero numbers L, T , The two equations show how the measures μ v and μ a are affected by a change of units according to u → L −1 · u and u t → T −1 · u t . Reasoning similarly [6, pp. 128-130], Fourier pointed out that quantity terms can be equal or combined by addition or subtraction only if they agree with respect to each exposant de dimension, having identical patterns of exponents in expressions such as LT −1 , LT −2 or L 2 , since otherwise the validity of numerical equations corresponding to quantity equations would depend on an arbitrary choice of units. He thus introduced the principle of dimensional homogeneity for equations that relate quantities.
Note that if q = μ · u then M · q = M · (μ · u) = Mμ · u, so (M · q) /u = Mμ = M (q/u). Thus, in a sense turning Fourier's argument around, we also have These equations show how μ v and μ a are affected when quantities change according to → L · , t → T · t. For any fixed units u and u t , we can express this as where Φ and Γ are the quantity-valued functions given by Φ ( , t) = F ( /u , t/u t ) · u v and Γ ( , w) = G ( /u , w/u ) · u a , respectively; note that u v and u a are also fixed since they depend on u and u t .
The homogeneity properties of Φ and Γ suggest that we write Φ ( , t) as α t −1 and Γ ( , w) as β w, where α and β are numerical constants. Generalizing this heuristic argument, we may introduce the idea that quantities of the same or different kinds can be multiplied and divided, suggesting that we can form arbitrary expressions of the form μ n i=1 q k i i , where μ is any number, q i are quantities and k i are integers, thus coming close to the quantity calculus set out below. Note, however, that Fourier did not actually define multiplication or division of quantities as such. This came later, with Lodge [3], Wallot [4] and others.
In retrospect, one may say that Fourier reinvented magnitudes as protoquantities and extended the range of applications. While Fourier reasoned in terms of multiplication and division of measures of quantities, he made a clear distinction between a quantity and its measure relative to a unit, this measure being a real number rather than an arithmos, he distinguished different kinds of quantities, and he considered new kinds of quantities such as temperatures and amounts of heat. Essential elements of a modern quantity calculus treating general quantities as mathematical objects (almost) as real as numbers were thus recognized early in the 19th century.
Subsequent progress in this area of mathematics has not been fast and straightforward, however. A Euclidean synthesis did not emerge; in his survey from 1994 de Boer concluded that "a satisfactory axiomatic foundation for the quantity calculus" had not yet been formulated [1]. One aim of this article is to provide such a foundation.
Gowers [14] points out that many mathematical objects are not defined directly by describing their essential properties, but indirectly by construction-definitions, specifying constructions that can be shown to have these properties. For example, an ordered pair (x, y) may be defined by a construction-definition as a set {x, {y}}; it can be shown that this construction has the required properties, namely that (x, y) = x , y if and only if x = x and y = y . Many contemporary formalizations of the notion of a quantity (e.g., [15,16]) use definitions relying on constructions, often defining quantities in terms of scalar-unit pairs, as did Maxwell. However, this is rather like defining a vector as a coordinates-basis pair rather than as an element of a vector space, the modern definition.
Although magnitudes are illustrated by line segments in the Elements, the notion of a magnitude is abstract and general. Remarkably, Euclid, following Eudoxus, dealt with this notion in a very modern way. While Euclid carefully defined other important objects such as points, lines and numbers in terms of inherent properties, there is no statement about what a magnitude "is". Instead, magnitudes are characterized by how they relate to other magnitudes through their roles in a system of magnitudes, to paraphrase Gowers [17].
In the same spirit, that of modern algebra, quantities are defined in this article simply as elements of a "quantity space". Thus, the focus is moved from individual quantities and operations on them to the systems to which the quantities belong, meaning that the notion of quantity calculus will give way to that of a quantity space. This article considers the notion of a quantity space introduced in [5] and developed further in [18]. (Further contemporary research on quantity calculus is found in [1,15,16,[19][20][21][22][23].) In the conceptual framework of universal algebra, a quantity space is just a special scalable monoid (X , * , (ω λ ) λ∈R , 1 X ), where X is the underlying set of the algebra, (X , * , 1 X ) is a monoid, R is a fixed ring and every ω λ is a unary operation on X . Writing * (x, y) as x y and denoting ω λ (x) by λ · x, we have The relation ∼ on a scalable monoid X defined by x ∼ y if and only if α · x = β · y for some α, β ∈ R is a congruence on X , so X is partitioned into corresponding equivalence classes. There is no global operation (x, y) → x + y defined on X , but within each equivalence class that contains a "unit element" addition of its elements is induced by the addition in R (see Sect. 2.6), and multiplication of equivalence classes is induced by the multiplication of elements of X (see Sect. 2.3).
Quantity spaces are to scalable monoids as vector spaces are to modules. Specifically, a quantity space Q is a commutative scalable monoid over a field, such that there exists a finite basis for Q, similar to a basis for a free abelian group. As noted, quantities are just elements of quantity spaces, and dimensions are equivalence classes in quantity spaces.
This article contains two main sections, namely Sect. 2 which deals with scalable monoids and Sect. 3 where scalable monoids are specialized to quantity spaces. There is also a final section with concluding remarks about quantity calculus.

Scalable monoids
A scalable monoid is a monoid whose elements can be multiplied by elements of a ring, and where multiplication in the monoid, multiplication in the ring, and multiplication of monoid elements by ring elements are compatible operations.
Scalable monoids are formally defined and compared to rings and modules in Sect. 2.1, and some basic facts about them are presented in Sect. 2.2. Sections 2.3 and 2.5 are concerned with congruences on scalable monoids and related notions such as commensurability, orbitoids, homomorphisms and quotient algebras, while direct and tensor products of scalable monoids are defined in Sect. 2.4. Scalable monoids with unit elements are investigated in Sects. 2.6 and 2.7. In particular, addition of elements in the same equivalence class is defined, and coherent systems of unit elements are discussed.

Mathematical background, main definition and simple examples
A unital associative algebra X over a (unital, associative but not necessarily commutative 1 ) ring R can be defined as a set, also denoted X , with three operations: (1) addition of elements of X , a binary operation + : (x, y) → x + y on X such that X equipped with + is an abelian group; (2) multiplication of elements of X , a binary operation * : (x, y) → x y on X such that X equipped with * is a monoid; (2) scalar multiplication of elements of X by elements of R, a monoid action (α, There are identities specifying a link between each pair of operations: (a) addition and multiplication of elements of X are linked by the distributive laws x (y + z) = x y + xz and (x + y) z = xz + yz; (b) addition of elements of X or R and scalar multiplication of elements of X by elements of R are linked by the distributive laws (c) multiplication of elements of X and scalar multiplication of elements of X by elements of R are linked by the homogeneity condition α · x y = (α · x) y and Related algebraic structures can be obtained from unital associative algebras by removing one of the operations (1)-(3) and hence the links between the removed operation and the two others. Two cases are very familiar: a ring has only addition and multiplication of elements of X , linked as described in (a), and a (left) module has only addition of elements of X and scalar multiplication of elements of X by elements of R, linked as described in (b). The question arises whether it would be meaningful and useful to define an "algebra without an additive group", with only multiplication of elements of X and scalar multiplication of elements of X by elements of R, linked as described in (c). The answer is affirmative. It turns out that this notion, a "scalable monoid", formally related to rings and in particular modules, makes sense mathematically and is remarkably well suited for modeling systems of quantities. The ancient arithmos-megethos pair of notions receives a modern interpretation: while numbers can be formalized as elements of rings, typically fields, quantities can be formalized as elements of scalable monoids, specifically quantity spaces. Definition 2.1 Let R be a (unital, associative) ring. A scalable monoid over R is a monoid X equipped with a scaling action by the multiplicative monoid of R We denote the identity element of X by 1 X , and set x 0 = 1 X for any x ∈ X . An invertible element of a scalable monoid X is an element x ∈ X that has a (necessarily It is easy to verify that the trivial scaling action of a ring R on a monoid X defined by λ · x = x for all λ ∈ R and x ∈ X is indeed a scaling action according to Definition 2.1. We call a monoid equipped with a trivial scaling action a trivially scalable monoid. A scalable monoid of this kind is essentially just a monoid, since the operation (λ, x) → λ · x can be disregarded in this case.

Example 2.2
A trivial scalable monoid is a trivial monoid {1 X } with a trivial scaling action.

Example 2.3
Let M (n) be the multiplicative monoid of all n × n matrices with entries in R. Then M (n) is a scalable monoid over the corresponding matrix ring R (n), with the scaling action defined by A · X = (det A) X.

Example 2.4
Let R x 1 ; . . . ; x n denote the set of all monomials of the form where R is a commutative ring, λ ∈ R, x 1 , . . . , x n are uninterpreted symbols and k 1 , . . . , k n are non-negative integers. We can define the operations ( . . . ; x n equipped with these operations is a commutative scalable monoid over R.

Some basic facts about scalable monoids
A not necessarily commutative scalable monoid over a not necessarily commutative ring nevertheless exhibits certain commutativity properties as described in the following useful lemma.

Lemma 2.5 Let X be a scalable monoid over R.
For any x, y ∈ X and α, β ∈ R we have Proof By Definition 2.1, where the first identity is used in the proof of the second.
Since every monoid M has a unique identity element 1 M , the class of all monoids forms a variety of algebras with a binary operation * : (x, y) → x y, a nullary operation 1 M : () → 1 M and identities The class of all scalable monoids over a fixed ring R is a variety in addition equipped with a set of unary operations {ω λ | λ ∈ R}, derived from the scaling action ω in Definition 2.1 by setting ω λ (x) = λ · x for all λ ∈ R and x ∈ X , and with the additional identities The scalable monoids is thus a variety of algebras belonging to the class of all (R, 1)-magmas where X is a carrier set, * a binary operation, ω λ a unary operation and 1 X a nullary operation. The general definitions of subalgebras, homomorphisms and products of algebras in the theory of universal algebras apply to (R, 1)-magmas. In particular, a subalgebra of an (R, 1)-magma X is a subset Y of X such that 1 X ∈ Y and x y, λ·x ∈ Y for any x, y ∈ Y and λ ∈ R. Also, a homomorphism φ : Recall, furthermore, that varieties are closed under the operations of forming subalgebras, homomorphic images and products since the defining identities are replicated by these operations [24]. Thus, a subalgebra of a scalable monoid over R, a homomorphic image of a scalable monoid over R, and a direct product of scalable monoids over R are all scalable monoids over R. Results related to these and other constructions will be considered in the remainder of Sect. 2.

Commensurability classes and canonical quotients
In ancient Greek mathematics, the notions of a sum, difference or ratio of magnitudes did not apply to magnitudes of different kinds, so in particular these could not be commensurable in the Greek (Pythagorean) sense. Moreover, magnitudes of the same kind, for example, two lengths, could nevertheless be incommensurable. In this section, we introduce a seemingly more radical idea: quantities are of the same kind if and only if they are commensurable. Definition 2.6 Given a scalable monoid X over R, let ∼ be the relation on X such that x ∼ y if and only if α · x = β · y for some α, β ∈ R. We say that x and y are commensurable if and only if x ∼ y; otherwise x and y are incommensurable.
Let R · x denote the set {λ · x | λ ∈ R}, that is, the orbit of x ∈ X for the scaling action ω : R × X → X , and let ≈ denote the relation on X such that x ≈ y if and only if there is some t ∈ X such that x, y ∈ R · t. Note that ≈ is not an equivalence relation; it is reflexive since x = 1 · x ∈ R · x for all x ∈ X and symmetric by construction but not transitive, meaning that the orbits for ω may overlap. On the other hand, x ∼ y if and only if (R · x) ∩ (R · y) = ∅, and this relation is indeed transitive.

Proposition 2.7 The relation ∼ on a scalable monoid X over R is an equivalence relation.
Proof The relation ∼ is reflexive since 1 · x = 1 · x for all x ∈ X , symmetric by construction, and transitive because if α · x = β · y and γ · y = δ · z for some x, y, z ∈ X and α, β, γ , δ ∈ R then it follows from Lemma 2.5 that where γ α, βδ ∈ R. Definition 2.8 A commensurability class or orbitoid C is an equivalence class for ∼. The orbitoid that contains x is denoted by [x], and X /∼ denotes the set For example, the commensurability classes of R x 1 ; . . . ; x n in Example 2.4 are the sets λx k 1 1 . . . x k n n | λ ∈ R for fixed non-negative integers k 1 , . . . , k n .

Remark 2.9
The orbits corresponding to an action of a group G on a set X are precisely the equivalence classes given by the equivalence relation ∼ G defined by x ∼ G y if and only if α · x = y for some α ∈ G; we clearly have G · x = G · y if and only if x ∼ G y. Similarly, orbitoids -generalized orbits in X under a monoid action satisfying α · (β · x) = β · (α · x)-are given by the equivalence relation ∼ defined by x ∼ y if and only if α · x = β · y for some α, β ∈ R. One may say that orbitoids generalize orbits as ∼ generalizes ∼ G .
It is instructive to compare the present notion of commensurability with the classical We say that x and y are strongly commensurable if and only if x ≈ y; otherwise, x and y are said to be weakly incommensurable.
Incommensurability of magnitudes in the Pythagorean sense obviously corresponds to weak incommensurability, so it is implied by, but does not imply, incommensurability in the present sense. Conversely, we have weakened the classical notion of commensurability here, at the same time making commensurability into an equivalence relation. The present concept of commensurability corresponds to the intuitive notion of magnitudes of the same kind, or the somewhat fuzzy notion of quantities of the same kind in modern theoretical metrology [25].

Remark 2.12
The mathematical quantities defined here may be interpreted as sizeproperties of certain objects or phenomena. Through one or more abstraction steps, concrete properties can be reduced to more abstract properties. The level of abstraction chosen affects the categorization of quantities into kinds of quantities. For example, it would seem that there are no scalars α, β such that α · x = β · y, where x is a planar angle and y a solid angle. A plane angle cannot be resized to a solid angle, or vice versa, so plane and solid angles would appear to be quantities of different kinds. However, this is a conclusion based on concrete properties of plane and solid angles. It is also possible to characterize them by commensurable abstract size-properties, so that they become quantities of the same kind. This is why both plane and solid angles are mainly regarded as "dimensionless" quantities (see also [18, pp. 8-12]).
It is often said that energy and torque are quantities of different kinds but with the same dimension. It should be noted, however, that the relation between energy and torque does not contradict the equivalence between dimensions and kinds of quantities proposed in this section, since a torque is not a quantity as defined here but a so-called vector quantity.
So far, we have regarded ∼ as an equivalence relation, but it turns out that more can be said.

Proposition 2.13
Let X be a scalable monoid over R. The relation ∼ is a congruence on X with regard to the operations (x, y) → x y and (λ, x) → λ · x.
In view of Proposition 2.13, we can define operations on X /∼ as follows. Given these definitions, X /∼ is an (R, 1)-magma and the surjective function so φ is a homomorphism of (R, 1)-magmas and thus of scalable monoids.

Proposition 2.15
If X is a scalable monoid over R then X /∼ is a scalable monoid over R, and the function is a surjective homomorphism of scalable monoids.
We call X /∼ the canonical quotient of X .

Proposition 2.16
If X is a scalable monoid then X /∼ is a trivially scalable monoid.

Direct and tensor products of scalable monoids
Consider an (R, 1)-magma where X and Y denote the underlying sets of two scalable monoids X and Y over R, * is a binary operation given by (x 1 , y 1 ) (x 2 , y 2 ) = (x 1 x 2 , y 1 y 2 ), where x 1 , x 2 ∈ X and y 1 , y 2 ∈ Y , each ω λ is a unary operation given by ω λ (x, y) = λ·(x, y) = (λ · x, λ · y), where x ∈ X and y ∈ Y , and 1 X ×Y is a nullary operation given by 1 X ×Y = (1 X , 1 Y ). Straight-forward calculations (or the HSP theorem [24]) show that this (R, 1)-magma, likewise denoted X × Y , is a scalable monoid over R. We call X × Y the direct product of X and Y . The direct product of scalable monoids is a generic product, applicable to any universal algebra. Another kind of product, which exploits the fact that (λ · x) y = λ · x y = x (λ · y) in scalable monoids, namely the tensor product, turns out to be more useful in many cases.

Definition 2.19
Let X and Y be scalable monoids, let x ⊗ y denote the equivalence class equipped with the operations given by We call X ⊗ Y the tensor product of X and Y . Proposition 2.20 Let X and Y be scalable monoids over R, x ∈ X and y ∈ X . Then (λ · x) ⊗ y = x ⊗ (λ · y) for every λ ∈ R.

Proposition 2.21 Let X and Y be scalable monoids over R. Then X ⊗ Y is a scalable monoid over R.
Proof X ⊗ Y is a monoid since It follows that if X , Y , Z are scalable monoids over R then (X ⊗ Y ) ⊗ Z and X ⊗ (Y ⊗ Z ) are scalable monoids over R: the tensor product of X ⊗ Y and Z in the first case and of X and Y ⊗ Z in the second case. It can also be shown that (X ⊗ Y )⊗ Z and X ⊗ (Y ⊗ Z ) are isomorphic scalable monoids.
The tensor product can be used to "glue" scalable monoids together in a natural way so as to combine them into more inclusive scalable monoids [22]. For example, R x; y is isomorphic to the tensor product R x ⊗ R y but not to the direct product R x × R y .

Quotients of scalable monoids by normal submonoids
In a monoid we have x (yz) = (x y) z and 1 X x = x = x1 X , so a submonoid M of a scalable monoid X can act as a monoid on X by left or right multiplication. In particular, we can define an action π : M × X → X by setting π (m, x) = mx for any m ∈ M and x ∈ X . This action can be used to define further notions in the same way that ∼, [x] and X /∼ were defined in terms of the scaling action ω : R × X → X .

Definition 2.22
Let X be a scalable monoid and M a submonoid of X . Then ∼ M is the relation on X such that x ∼ M y if and only if mx = ny for some m, n ∈ M.
A normal submonoid of a scalable monoid X is a submonoid M of X such that xM = Mx for every x ∈ X . It is clear that if M is a central submonoid of X , that is, if every element of M commutes with every element of X , then M is normal, and every submonoid of a commutative scalable monoid is normal.

Proposition 2.23 If X is a scalable monoid and M a normal submonoid of X then ∼ M is an equivalence relation on X .
Proof The relation ∼ M is reflexive since 1 X x ∼ M 1 X x for all x ∈ X , symmetric by construction, and transitive because if mx = ny and m y = n z for x, y, z ∈ X and m, n, m , n ∈ M then there is some n 0 ∈ M such that m mx = m ny = n 0 m y = n 0 n z, where m m, n 0 n ∈ M.
Proof If if α · x = β · y for some α, β ∈ R then This implies the assertion since α · 1 X , β · 1 X ∈ M. Proposition 2.30 Let X be a scalable monoid over R and x, y ∈ X . Then R · 1 X is a normal scalable submonoid of X , and x ∼ R·1 X y implies x ∼ y.
It follows from Propositions 2.29 and 2.30 that x ∼ M y generalizes x ∼ y.

Corollary 2.31 Let X be a scalable monoid over R and x, y ∈ X . Then R · 1 X is a normal scalable submonoid of X , and x ∼ R·1 X y if and only if x ∼ y.
for any κ ∈ R, so R x 1 ; . . . ; x n /∼ and R x 1 ; . . . ; x n / (R · 1 X ) are isomorphic monoids.

A non-trivial orbitoid with a unit element is a free module of rank 1
Recall the principle that magnitudes of the same kind can be added and subtracted, whereas magnitudes of different kinds cannot be combined by these operations. Also recall the idea that a quantity can be represented by a "unit" and a number (measure) specifying "[how many] times the [unit] is to be taken in order to make up" that quantity [2, p. 41]. As shown below, there is a connection between these two notions. Specifically, it may happen that R · u ⊇ [u] for some u ∈ [u], and if in addition a natural uniqueness condition is satisfied we may regard u as a unit of measurement for [u]. If such a unit exists then a sum of elements of [u] can be defined by the construction described in Definition 2.35 below.

Definition 2.32
Let C be an orbitoid in a scalable monoid over R. A generating element for C is some u ∈ C such that for every x ∈ C there is some λ ∈ R such that x = λ · u. A unit element for C is a generating element u for C such that if λ · u = λ · u then λ = λ .
By this definition, if u is a generating element for C = [u] then R · u ⊇ C. On the other hand, λ · u ∼ u for any λ ∈ R, so λ · u ∈ [u] for any λ ∈ R, so R · u ⊆ [u]. Thus, actually R · u = C.
We now need to consider zero elements in scalable monoids.
We call 0 C the zero element of C; note that distinct orbitoids have distinct zero It is clear that λ · 0 C = 0 C for all λ ∈ R, and that 0 [x] y = 0 [xy] and y0 [x] = 0 [yx] for all x, y ∈ X .
We now turn to a lemma and a definition leading to Proposition 2.36.
Proof As u ∈ C, there is a unique τ ∈ R such that u = τ · u. Thus, Hence, the sum of two elements of a scalable monoid can be defined as follows.

Definition 2.35
Let X be a scalable monoid over R, and let u be a unit element for C ∈ X /∼. If x = ρ · u and y = σ · u, where ρ, σ ∈ R, we set The sum x + y is given by Definition 2.35 if and only if x and y are commensurable and their orbitoid has a unit element. This suggests again that the concept of commensurability introduced in Definition 2.32 can be used to define the ancient Greek notion of magnitudes of the same kind, and to clarify the modern notion of quantities of the same kind.
It follows immediately from Definition 2.35 that for all x, y, z ∈ C, and that for any x ∈ C since 0 C = 0 · u.
If x = ρ · u so that λ · x = λρ · u and κ · x = κρ · u then and if x = ρ · u and y = σ · u so that λ · x = λρ · u and λ · y = λσ · u then A unital ring R has a unique additive inverse −1 of 1 ∈ R, and we set for all x ∈ X . If C has a unit element u and x = ρ · u for some ρ ∈ R then −x = (−1) · (ρ · u) = (−ρ) · u, and using this fact it is easy to verify that As usual, we may write x + (−y) as x − y, and thus While a trivial orbitoid is a zero module {0 C } with 0 C + 0 C = 0 C and λ · 0 C = 0 C for all λ ∈ R, a non-trivial orbitoid with a unit element is a well-behaved module.

Proposition 2.36
Let X be a scalable monoid over R. If C ∈ X /∼ is a non-trivial orbitoid with a unit element then R is a non-trivial commutative ring, and C, with appropriate definitions of x + y and λ · x, is a free module of rank 1 over R.
Proof Let u be a unit element for C. If 0 C = x ∈ C, 0 C = λ · u and x = κ · u for some λ, κ ∈ R then λ = κ, so R is non-trivial. We also have αβ · u = βα · u for any α, β ∈ R by Lemma 2.5, so αβ = βα since u is a unit element.
We have seen that C is a module with addition given by Definition 2.35 and scalar multiplication inherited from the scalar multiplication in X . Also, if u is a unit element for C then {u} is a basis for C, and R has the invariant basis number property since it is non-trivial and commutative [26].
Thus, if every orbitoid C ∈ X /∼ contains a non-zero unit element for C then X is the union of disjoint isomorphic free modules of rank 1 over a non-trivial commutative ring, a result that may be compared to definitions of systems of quantities in terms of unions of one-dimensional vector spaces by Quade [27] and Raposo [22].

Scalable monoids with sets of unit elements
In this section, we build on the discussion in the two previous sections about unit elements and quotients of scalable monoids by normal monoids.

Definition 2.37
A dense set of elements of a scalable monoid X is a set U of elements of X such that for every x ∈ X there is some u ∈ U such that u ∼ x. A sparse set of elements of X is a set U of elements of X such that u ∼ v implies u = v for any u, v ∈ U . A closed set of elements of X is a set U of elements of X such that if u, v ∈ U then uv ∈ U .
We call a (dense) sparse set of unit elements of X a (complete) system of unit elements for X .

Definition 2.38
A distributive scalable monoid X is a scalable monoid such that for all A, B ∈ X /∼ we have for all x, y ∈ A and all z ∈ B.

Proposition 2.39 Let X be a scalable monoid. If X is equipped with a dense closed set of unit elements U then X is a distributive scalable monoid.
Proof using the fact that uv and vu are unit elements since U is closed.
We also define a natural notion which is fundamental in metrology.

Definition 2.40
A coherent system of unit elements for X is a submonoid of X which is a complete system of unit elements for X .
Recall that if T is a normal submonoid of a scalable monoid X then X /T is a scalable monoid by Proposition 2.27. It is proved in [28, Appendix A] that if S ⊇ T is a coherent system of unit elements for X then S/T is a coherent system of unit elements for X /T .

Quantity spaces
In this section, we specialize scalable monoids in order to obtain a mathematical model suitable for calculation with quantities, a quantity space.
The formal definition of a quantity space is given in Sect. 3.1, and some basic facts about quantity spaces are presented in Sect. 3.2. Coherent systems of unit quantities for quantity spaces are discussed in Sect. 3.3. The notion of a measure of a quantity is formally defined in Sect. 3.4, and ways in which measures serve as proxies for quantities are described. In Sect. 3.5, we show that the monoid of dimensions Q/∼ corresponding to a quantity space Q is a free abelian group and derive some related results.

Canonical construction and main definition
It is possible to give an abstract definition of scalable monoids of the form R x 1 ; . . . ; x n (Example 2.4). Let X be a commutative scalable monoid over a commutative ring R. A finite scalable-monoid basis for X is a finite set {e 1 , . . . , e n } of elements of X such that every x ∈ X has a unique expansion where μ ∈ R and k i are non-negative integers. R x 1 ; . . . ; x n is clearly a commutative scalable monoid over a commutative ring with a finite scalable-monoid basis 1x where δ i j is the Kronecker delta, and it can be shown that conversely every commutative scalable monoid over a commutative ring with a finite scalable-monoid basis is isomorphic to some R x 1 ; . . . ; x n . Now, consider instead the set K x 1 , x −1 1 ; . . . ; x n , x −1 n of all Laurent monomials of the form λx k 1 1 . . . x k n n , where λ ∈ K , K is a field, x 1 , . . . , x n are uninterpreted symbols and k 1 , . . . , k n are integers, together with essentially the same operations as in Example 2.4, namely is likewise a scalable monoid, which can again be characterized abstractly. Definition 3.1 Let Q be a commutative scalable monoid over a field K . A finite quantity-space basis for Q is a finite set {e 1 , . . . , e n } of invertible elements of Q such that every x ∈ Q has a unique expansion where μ ∈ K and k i are integers.
It is easy to show that any commutative scalable monoid Q over a field, such that there exists a finite quantity-space basis for Q, can be represented by some [18]. On the other hand, we have the following abstract characterization of this kind of scalable monoid, corresponding to a finitely generated free abelian group and well suited for applications in theoretical metrology, dimensional analysis, etc. Definition 3.2 A finitely generated quantity space is a commutative scalable monoid Q over a field, such that there exists a finite quantity-space basis for Q.
Although finitely generated quantity spaces can be readily generalized to quantity spaces with infinite bases, only the finite case will be considered here. Below, "basis" and "quantity space" will be understood to mean "finite quantity-space basis" and "finitely generated quantity space", respectively.
Elements of a quantity space are called quantities, unit elements are called unit quantities, and orbitoids in a quantity space are called dimensions.
Note that K x 1 ; . . . ; x n , where K is a field, is not a quantity space, so a commutative scalable monoid over a field is not necessarily a quantity space: the relationship between a scalable monoid and a quantity space is not as close as that between a module and a vector space.

Some basic properties of quantity spaces
Proof (1) Note that e 0 i = 1 Q for all e i . (2) This follows from Lemma 2.5 and the fact that Q is commutative. (3) μ −1 ∈ K as μ = 0, and e 1 , . . . , e n ∈ Q are invertible by definition, so (1) and (2).

Proposition 3.4
Let Q be a quantity space over K with a basis {e 1 , . . . e n } and x = μ · n i=1 e k i i . Then the following conditions are equivalent: (1) x is a non-zero quantity; (2) ⇐⇒ (3). If μ = 0 then x has an inverse by Proposition 3.3. Conversely, if μ = 0 then μν = 0 = 1 for all ν ∈ K , so x does not have an inverse ν · n i=1 e i i .
Thus, 1 Q is a non-zero quantity, and all elements of a basis are non-zero quantities. Also, it follows from Proposition 3.4 that Q has no zero divisors. (1) x ∼ y, or equivalently μ · n i=1 e k i i ∼ ν · n i=1 e i i ; (2) k i = i for i = 1, . . . , n; Proof Implications ( It follows immediately from Lemma 3.6 that if not k i = i for i = 1, . . . , n then x = y since x y; this is the essence of the principle of dimensional homogeneity formulated by Fourier [6].

Proposition 3.7 If Q is a quantity space then every non-zero u ∈ Q is a unit quantity for [u].
Proof Let {e 1 , . . . , e n } be a basis for Q and set u = μ · n i=1 e k i i , x = ν · n i=1 e i i . Then μ = 0 by Proposition 3.4, and if x ∼ u then μ · x = ν · u by Lemma 3.6, so e k i i , so λμ = λ μ since the expansion of λ · u is unique, so λ = λ since μ = 0.

Proposition 3.8 If Q is a quantity space then every C ∈ Q/∼ contains a non-zero unit quantity.
Proof If x = μ · n i=1 e k i i ∈ C then u = 1 · n i=1 e k i i is non-zero by Proposition 3.4 and u ∈ C by Lemma 3.6, so u is a unit quantity for C by Proposition 3.7. Proposition 3.9 Let Q be a quantity space over K . Then C ∈ Q/∼, with x + y and λ · x appropriately defined, is a one-dimensional vector space over K .
Proof C is a free module of rank 1 over the field K by Propositions 2.36 and 3.8.
of Q is a coherent system of unit quantities for Q.
Proof By Proposition 3.4, all elements of U are non-zero and hence unit quantities by Proposition 3.7. Also, U is dense in Q since it follows from Lemma 3.6, so u = v, meaning that U is sparse in Q. It remains to prove that U is a monoid. Clearly, 1 Q ∈ U since 1 Q = 1 · n i=1 e 0 i , and we have In other words, every basis can be extended to a coherent system of unit quantities, consisting of quantities that are expressed as products of basis quantities and their inverses. As a direct consequence, we have the following result.

Proposition 3.11 If Q is a quantity space over K then Q is distributive.
Proof The assertion follows from Propositions 3.10 and 2.39.

Measures of quantities
Definition 3.12 Let Q be a quantity space over K with a basis E = {e 1 , . . . , e n }. The uniquely determined scalar μ ∈ K in the expansion is called the measure of x relative to E and will be denoted by μ E (x).
For example, 1 Q = 1 · n i=1 e 0 i for any E, so we have the following simple but useful fact.

Proposition 3.13
If Q is a quantity space over K then μ E 1 Q = 1 for any basis E for Q.
Relative to a fixed basis, measures of quantities can be used as proxies for the quantities themselves. Proposition 3.14 Let Q be a quantity space over K with a basis E = {e 1 , . . . , e n }. Then for all x, y ∈ X such that x ∼ y.
Hence, x has an expansion in terms of E . To prove uniqueness, assume that Changing this expansion in terms of E to an expansion in terms of E = λ −1 1 · (λ 1 · e 1 ) , . . . , λ −1 n · (λ n · e n ) gives so μ = μ E (x) and i = k i for i = 1, . . . , n by the uniqueness of the expansion of x in terms of E = E .
In general, the measure of a quantity thus depends on a choice of basis, but there is an important exception to this rule. Proposition 3.16 Let Q be a quantity space over K . For every x ∈ 1 Q , μ E (x) does not depend on E.
Proof 1 Q is a unit quantity for 1 Q by Proposition 3.7, so there is a unique λ ∈ K such that x = λ · 1 Q , so μ E (x) = λ μ E 1 Q = λ for any basis E for Q by Propositions 3.13 and 3.14(3).

Remark 3.17
The π theorem in dimensional analysis depends on this result [29]. It is common to refer to any x ∈ 1 Q as a "dimensionless quantity", although x is not really dimensionless-it belongs to, or "has", the dimension 1 Q . Also, many authors (e.g., [4,19,23]) identify "dimensionless quantities" with numbers, but Proposition 3.16 does not fully justify this identification. A "dimensionless quantity" does not correspond to a unique number, but to a number that depends on the choice of a quantity unit for 1 Q . For example, plane angles can be measured in both radians and degrees. However, if we have a coherent system of units U then 1 Q contains exactly one unit 1 Q since U is a submonoid of Q and u ∼ 1 Q implies u = 1 Q . Also, by Proposition 3.10 each choice of basis for Q-that is, each choice of so-called base units [25]-gives rise to a coherent system of units. Note that for a plane angle 1 Q corresponds to the radian.

Q/∼ is a free abelian group
In this section, we show that Q/∼ regarded as a monoid has additional properties derived from the quantity space Q. Below, letx be given byx i is the expansion of x ∈ Q relative to a basis for Q. Note that, irrespective of the choice of basis,x is a non-zero quantity by Proposition 3.4 and such thatx ∼ x by Lemma 3.6.

Proposition 3.18
If Q is a quantity space then Q/∼ is an abelian group.
Recall that a basis for a finitely generated abelian group G is a set {ε 1 , . . . , ε n } of elements of G such that every x ∈ G has a unique expansion Proof The unique expansions of e i , e j ∈ E relative to E are Hence, if e i = e j so that i = j then [e i ] = e j by Lemma 3. 6 be an arbitrary dimension in Q/∼. As E is a basis for Q, we have x = μ · n i=1 e k i i for some μ ∈ K and some integers k 1 , . . . , k n , so A (finitely generated) abelian group for which there exists a basis is said to be free abelian (of finite rank). Hence, corresponding to the fact that if X is a scalable monoid then X /∼ is a monoid, we have the following much stronger result.

Proposition 3.20 If Q is a quantity space then Q/∼ is a free abelian group of finite rank.
Recall that any two bases for a free abelian group G have the same cardinality, the rank of G. Proposition 3.19 thus leads to an analogue of the dimension theorem for finite-dimensional vector spaces. A quantity space with bases of cardinality n is said to be of rank n.

Example 3.22
The dimensions corresponding to base quantities in the International System of Units (SI) [30], such as the dimensions of length, time and mass, denoted L, T and M, respectively, are elements of a basis for some free abelian group Q/ ∼. For example, {L, T, M} is a basis for Q/ ∼, where Q is a quantity space for classical mechanics. This is not the only possible basis, however. For example, {L, T, F}, where F = MLT −2 , is another three-element basis for Q/ ∼, and another possible set of base dimensions for classical mechanics.
Let us consider quantity spaces Q and Q over K with bases E = {e 1 , . . . , e n } and E = e 1 , . . . , e n . It is easy to verify that a bijection φ : E → E can be extended to an isomorphism φ * : Q → Q by setting φ * μ · n i=1 e k i i = μ · n i=1 φ (e i ) k i . Conversely, if φ * : Q → Q is an isomorphism then {φ * (e 1 ) , . . . , φ * (e n )} is clearly a basis for Q of the same cardinality as E. These observations lead to the following classification theorem, similar to a theorem in linear algebra.

Proposition 3.23
Quantity spaces over the same field are isomorphic if and only if they are of the same rank (cf. [22]).
There is a reciprocal connection between bases for Q and bases for Q/ ∼.

Proposition 3.24
Let Q be a quantity space, and let E = {e 1 , . . . , e n } be a basis for Q/∼. Then there is a subset E = {e 1 , . . . , e n } of Q such that e i ∈ e i and E is a basis for Q.
Proof We can choose a function ψ : E → {ψ (e 1 ) , . . . , ψ (e n )} such that we have 0 e i = ψ (e i ) ∈ e i for all e i . This is a surjective function, and e i = e j implies e i ∩ e j = ∅, so ψ is injective as well and hence a bijection. For convenience, we write ψ (e i ) as e i . Each e i is invertible by Proposition 3.4.
Let x be an arbitrary quantity in Q. As E is a basis for Q/∼, we have for some integers k 1 , . . . , k n , and as e i = 0 e i for each e i , n i=1 e k i i is non-zero and thus a unit quantity for [x] by Proposition 3.7. Hence, there exists a unique μ ∈ K for n i=1 e k i i such that [e i ] i , so i = k i for i = 1, . . . , n, since E is a basis for Q/∼, so ν = μ.
We can now extend to quantity spaces the theorem that a subgroup of a free abelian group is free abelian, using this fact.

Proposition 3.25 If a subalgebra Q of a quantity space Q regarded as a scalable monoid contains all inverses of elements of Q then Q is a quantity space.
Proof First note that Q /∼ is a submonoid of Q/∼. Also, recall from the proof of Proposition 3.18 that x −1 = [x] −1 so ifx ∈ Q impliesx −1 ∈ Q then [x] ∈ Q /∼ implies [x] −1 ∈ Q /∼ since x ∈ Q impliesx ∈ Q . Hence, Q /∼ is a subgroup of Q/∼, so Q /∼ is a free abelian group with a basis E corresponding to a basis E for Q by Proposition 3.24.
This result is analogous also to the simple fact that a submodule of a vector space is a vector space, so we have found yet another similarity between free abelian groups, quantity spaces and vector spaces.

Quantity calculus and its foundation
In his survey from 1994, de Boer [1] concluded that quantity calculus had not yet found its final form. He listed some fundamental rules that calculation with quantities must obey in a definitive version of quantity calculus. All these rules can be derived from the theory of scalable monoids and quantity spaces presented here. The list below includes references to de Boer's rules and relevant definitions or results from this article. (Note that de Boer makes a distinction between dimensions and equivalence classes of quantities of the same kind, whereas these two notions coincide in the present theory.) of kind K then uu is the unit of kind K K . A set of units satisfying this condition is said to be coherent. We can also derive two fundamental rules not considered by de Boer in [1]. (i) (Corollary 2.11). λ · q is a quantity of the same kind as q.
(ii) (Proposition 3.11). If q is a quantity and r , s are quantities of the same kind then q (r + s) = qr + qs. The slow progress seen in the history of quantity calculus can be partly attributed to a peculiarity of the mathematics required. Quantity calculus is special in that only quantities of the same kind can be added. Thus, addition of quantities, unlike addition of numbers or vectors, would be a partial operation, and the notion of quantities of the same kind would have to be defined formally, in purely mathematical terms. In addition, this should preferably be achieved by inference from axioms and definitions rather than by using a construction where sets of quantities of the same kind are given as a part of the structure of the space of quantities. In the present approach to quantity calculus, these challenges are addressed as described in Sects. 2.2-2.3 and 2.6, leading up to the pivotal Proposition 2.36, according to which elements of the same kind typically constitute a free module of rank 1, that is, a one-dimensional vector space, when the scalable monoid is specialized to a quantity space.
As we have seen, this innovative approach required some input from abstract algebra, namely a new algebraic structure: scalable monoids. These would seem to fill a void between modules and rings in the area of two-operator algebraic structures. Apart from providing a rigorous foundation for quantity calculus and related areas of application, the definitions and results presented in this article may stimulate further mathematical research on scalable monoids.