Mobility spaces and their geodesic paths
aa r X i v : . [ m a t h . G M ] J a n MOBILITY SPACES AND THEIR GEODESIC PATHS
J. P. FATELO AND N. MARTINS-FERREIRA
Abstract.
We introduce an algebraic system which can be usedas a model for spaces with geodesic paths between any two of theirpoints. This new algebraic structure is based on the notion of mo-bility algebra which has recently been introduced as a model forthe unit interval of real numbers. Mobility algebras consist on a set A together with three constants and a ternary operation. In thecase of the closed unit interval A = [0 , p ( x, y, z ) = x − yx + yz . Amobility space is a set X together with a map q : X × A × X → X with the meaning that q ( x, t, y ) indicates the position of a particlemoving from point x to point y at the instant t ∈ A , along a ge-odesic path within the space X . A mobility space is thus definedwith respect to a mobility algebra, in the same way as a module isdefined over a ring. We introduce the axioms for mobility spaces,investigate the main properties and give examples. We also estab-lish the connection between the algebraic context and the one ofspaces with geodesic paths. The connection with affine spaces isbriefly mentioned. January 13, 2020; 01:36:581.
Introduction
The purpose of this work is to introduce an algebraic system whichcan be used to model spaces with geodesics. The main idea stems fromthe interplay between algebra and geometry. In Euclidean geometrythe notion of affine space is well suited for this purpose. Indeed, inan affine space we have scalar multiplication, addition and subtractionand so it is possible to parametrize, for any instant t ∈ [0 , x and y with the formula (1 − t ) x + ty . Such a lineis clearly a geodesic path from x to y . In general terms, we may usean operation q = q ( x, t, y ) to indicate the position, at an instant t , of aparticle moving in space from a point x to a point y . If the particle ismoving along a geodesic path then this operation must certainly verify Mathematics Subject Classification.
Primary 08A99, 03G99; Secondary20N99, 08C15.
Key words and phrases.
Mobility algebra, mobility space, affine space, affinemobility space, unit interval, ternary operation, geodesic path.This work is supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT)and Centro2020 through the Project references: UID/Multi/04044/2019; PAMI -ROTEIRO/0328/2013 (N º some conditions. The aim of this project is to present an algebraicstructure, ( X, q ), with axioms that are verified by any operation q representing a geodesic path in a space between any two of its points.First results concerning this investigation were presented in [5] wherea binary operation, obtained by fixing t to a value that positions theparticle at half way from x to y , is studied. The whole movement of aparticle on a geodesic path is captured when the variable t is allowed torange over a set of values, of which the unit interval is the most naturalchoice. The investigation of those structures and properties relevantto our study led us to the discovery of a new algebraic structure thatwas called mobi algebra [7]. A mobility algebra (or mobi algebra),besides being a suitable algebraic model to the unit interval, offers aninteresting comparison with rings. A slogan may be used to illustratethat comparison: a mobi algebra is to the unit interval in the same wayas a ring is to the set of reals . A mobi algebra is an algebraic system( A, p, , ⁄ ,
1) consisting of a set together with a ternary operation p and three constants. Moreover, every ring in which 2 is invertible hasa mobi algebra structure, while a mobi algebra in which the element ⁄ is invertible (as defined in [7]) has a ring structure.Following the analogy with rings, and extending it to modules over aring, we have considered a new structure, called mobility space (mobispace for short). If A is a mobi (or mobility) algebra then a mobi space,say ( X, q ), is defined over a mobi algebra in the sense that q = q ( x, t, y )operates on x, y ∈ X and t ∈ A . The axioms defining a mobi spaceare similar to the ones defining a mobi algebra and examples show thatthese axioms are indeed appropriate to capture the features of spaceswith geodesic paths between any two of their points.This paper is organized as follows. The notion of mobi algebra [7] isrecalled together with its derived operations (Section 2). Two impor-tant new sorts of examples are added to the list presented in [7]. A firstsort of examples is constructed (Proposition 2.1) from a midpoint alge-bra by considering a suitable subset of its endomorphisms (Definition2.3). Another sort of examples is generated by considering a unitaryring in which the element 2 is invertible. As a particular case, the setof endomorphisms of an abelian group is canonically equipped with aunitary ring structure and hence, if the map x x is invertible, thenit gives rise to a mobi algebra structure on its set of endomorphisms(Proposition 2.2). These examples will be used to characterize affinemobi spaces which include, among other things, all vector spaces overa field and will be considered in a sequel of this work. Nonetheless,our last section is dedicated to a thoroughly comparison between affinemobi spaces and modules over a ring (Theorem 6.1 and Theorem 6.2).In addition, some details are given in the form of examples in subsection4.3. Section 3 is totally devoted to the definition of mobi space and a shortlist of its properties that will be needed in this paper. Further studiesare postponed to a future work, namely the transformations betweensuch structures, and the fact that the category of mobi spaces is aweakly Mal’tsev category [5, 12, 13]. Or its comparison with the workof A. Kock on affine connections with neighbouring relations [8, 9, 10]and its relation to the work of Buseman [1, 2] on spaces with uniquegeodesic paths.Section 4 is dedicated to an exhaustive list of worked examples andcounter examples that have been used for testing the strength of ouraxioms. These include simple examples (Example 4.1), more com-plex examples (Example 4.2), the particular case of affine mobi spaces(which are comparable to euclidean geometry) and some examples witha physical interpretation (Examples 4.5 and 4.7). In particular, we ob-serve that even for other type of physical equations, rather than thosefor geodesics, there is a procedure which can be followed to includephysical phenomenon into a mobility space. The procedure is basedon the simple trick of considering time as a geometrical dimension (seeExample 4.7 and Corollary 5.6).In Section 5 we study the main examples which arise as the solutionto the equation for geodesics and extend that result to cover a widerange of cases in which geodesics are not uniquely determined. This isdone via the process of identification spaces. In the end we have addeda section with a comparison between affine mobi spaces and modulesover a unitary ring in which the element 2 is invertible.2.
Mobi algebra
In this section we briefly recall the notion of mobi algebra which wasintroduced in [7] and some of its basic properties. A mobi algebra con-sists of a set equipped with three constants and one ternary operation.The unit interval [0 , , 1 and the formula p ( a, b, c ) = a + bc − ba has always been the example of a mobi algebrawhich have driven our intuition. Nevertheless, several other examplesof very different nature were presented in [7]. At the end of this sectionwe add two new important classes to the list of examples. As we willsee in the next section (while introducing the notion of mobi space) amobility algebra plays the role of scalars (for a mobility space) in thesame way as a ring (or a field) models the scalars for a module (or avector space) over the base ring (or field).In order to have an intuitive interpretation of its axioms we may con-sider a mobi algebra as a mobi space over itself and use the geometricintuition provided in section 3. Namely, that the operation p ( x, t, y )is the position of a particle moving from a point x to a point y at aninstant t while following a geodesic path. J. P. FATELO AND N. MARTINS-FERREIRA
Definition 2.1. [7]
A mobi algebra is a system ( A, p, , ⁄ , , in which A is a set, p is a ternary operation and , ⁄ and are elements of A,that satisfies the following axioms: (A1) p (1 , ⁄ ,
0) = ⁄ (A2) p (0 , a,
1) = a (A3) p ( a, b, a ) = a (A4) p ( a, , b ) = a (A5) p ( a, , b ) = b (A6) p ( a, ⁄ , b ) = p ( a, ⁄ , b ) = ⇒ b = b (A7) p ( a, p ( c , c , c ) , b ) = p ( p ( a, c , b ) , c , p ( a, c , b )) (A8) p ( p ( a , c, b ) , ⁄ , p ( a , c, b )) = p ( p ( a , ⁄ , a ) , c, p ( b , ⁄ , b )) . Some properties of mobi algebras can be suitably expressed in termsof a unary operation ”()” and binary operations ” · ”, ” ◦ ” and ” ⊕ ”defined as follows (see [7] for more details). Definition 2.2.
Let ( A, p, , ⁄ , be a mobi algebra. We define: a = p (1 , a,
0) (1) a · b = p (0 , a, b ) (2) a ⊕ b = p ( a, ⁄ , b ) (3) a ◦ b = p ( a, b, . (4)We recall a list of some properties of a mobi algebra. If ( A, p, , ⁄ , ⁄ = ⁄ (5) a · ⁄ = ⁄ · a = 0 ⊕ a (6) ⁄ · a = ⁄ · a ′ ⇒ a = a ′ (7) p ( a, ⁄ , a ) = ⁄ (8) a = a ⇒ a = ⁄ (9) p ( a, b, c ) = p ( a, b, c ) (10) p ( c, b, a ) = p ( a, b, c ) (11) a ◦ b = b · a (12) ⁄ · p ( a, b, c ) = ( b · a ) ⊕ ( b · c ) . (13)A long list of examples is provided in [7]. The following exampleshows that any non-empty midpoint algebra ( X, ⊕ ), with a chosen el-ement e ∈ X in it, gives rise to a mobi algebra structure on a suitablesubset of endomaps of X . This example will be crucial for the charac-terization of affine mobi spaces (see Section 4.3). Let us recall that a midpoint algebra [3] consists of a set X and abinary operation ⊕ satisfying the following axioms:(idempotency) x ⊕ x = x (commutativity) x ⊕ y = y ⊕ x (cancellation) x ⊕ y = x ′ ⊕ y ⇒ x = x ′ (mediality) ( x ⊕ y ) ⊕ ( z ⊕ w ) = ( x ⊕ z ) ⊕ ( y ⊕ w ) . It is remarkable that a certain subset of endomaps of X can always begiven the structure of a mobi algebra. Definition 2.3.
Let ( X, ⊕ ) be a non-empty midpoint algebra and e ∈ X a fixed chosen element in X . We denote by End ⊕ e ( X ) the set of en-domaps of X consisting of those maps g : X → X satisfying the follow-ing conditions:(i) g ( x ⊕ y ) = g ( x ) ⊕ g ( y ) , for all x, y ∈ X ,(ii) g ( e ) = e (iii) for all x ∈ X , there exists ¯ g ( x ) ∈ X such that ¯ g ( x ) ⊕ g ( x ) = e ⊕ x (iv) for all x, y ∈ X , there exists ˜ g ( x, y ) ∈ X such that ¯ g ( x ) ⊕ g ( y ) = e ⊕ ˜ g ( x, y )Let us observe that when g ∈ End ⊕ e ( X ) then ¯ g , defined as in ( iii ),is also in End ⊕ e ( X ). Moreover, we have ¯¯ g = g and ˜¯ g ( x, y ) = ˜ g ( y, x ).An example is provided in subsection 4.3, where affine mobi spaces areconsidered. Proposition 2.1.
Let ( X, ⊕ ) be a non-empty midpoint algebra and fixan element e ∈ X . The system ( End ⊕ e ( X ) , p , , , ) , with p ( x ) = p f,g,h ( x ) = ˜ g ( f ( x ) , h ( x )) , ( x ) = e , ( x ) = e ⊕ x and ( x ) = x is amobi algebra.Proof. First we show that the operation p is well defined. To do thatwe observe that if g ∈ End ⊕ e ( X ) then ˜ g , defined as in ( iv ), is such that˜ g ( x ⊕ x , y ⊕ y ) = ˜ g ( x , y ) ⊕ ˜ g ( x , y ) (14)˜ g ( x, x ) = x (15)Now, given f, g, h ∈ End ⊕ e we have to show that p f,g,h , defined for every x ∈ X as ˜ g ( f ( x ) , h ( x )), belongs to End ⊕ e . The first two conditions ( i )and ( ii ) follow from the properties of ˜ g . Condition ( iii ) is satisfiedwith ¯ p f,g,h ( x ) = ˜ g ( ¯ f ( x ) , ¯ h ( x )), while condition ( iv ) is satisfied with˜ p f,g,h ( x, y ) = ˜ g ( ˜ f ( x, y ) , ˜ h ( x, y )). The fact that the axioms of a mobialgebra (Definition 2.1) are verified by p f,g,h ( x ) is then a consequence ofTheorem 6.2 from [7]. Indeed, the system ( End ⊕ e ( X ) , () , ⊕ , · , ) with( g )( x ) = g ( x ), ( f ⊕ g )( x ) = f ( x ) ⊕ g ( x ), ( f · g )( x ) = f ( g ( x )) and ( x ) = x is a IMM* algebra in the sense of Definition 6.1 from [7].Moreover, the equation ⊕ χ = ( g · f ) ⊕ ( g · h ) has a solution in End ⊕ e ( X ), namely χ ( x ) = ˜ g ( f ( x ) , h ( x )) for every x ∈ X . (cid:3) J. P. FATELO AND N. MARTINS-FERREIRA
As it was shown in [7], every unitary ring in which the element 2is invertible gives rise to a mobi algebra structure. A particular case,which will be used in section 4.3, is the ring of endomorphisms of anabelian group. Let ( X, + ,
0) be an abelian group. It is clear that themap x x + x is invertible if and only if for every x ∈ X , there exists y ∈ X with y + y = x . For obvious reasons the element y such that y + y = x will be denoted x . As usual, we denote the set of groupendomorphisms of X by End ( X ). Proposition 2.2.
Let ( X, + , be an abelian group in which the en-domorphism sending x to x + x is invertible. The system ( End ( X ) , p , , , ) , with p ( x ) = p f,g,h ( x ) = f ( x ) − gf ( x ) + gh ( x ) , ( x ) = 0 , ( x ) = x and ( x ) = x is a mobi algebra.Proof. The result follows directly from Theorem 7.2 in [7] since
End ( X )is a unitary ring with one half. (cid:3) In spite of these somehow more abstract examples, the reader iskindly reminded that it is useful to keep in mind that the most naturaland intuitive mobi algebra is the closed unit interval with A = [0 , , 1 and the operation p ( a, b, c ) = (1 − b ) a + bc .The mobi algebra, in this case, is interpreted as the set of scalars overwhich a mobi space is constructed.3. Mobi space
In this section we give the definition of a mobi space over a mobialgebra. Its main purpose is to serve as a model for spaces with ageodesic path connecting any two points. It is similar to a moduleover a ring in the sense that it has an associated mobi algebra whichbehaves as the set of scalars. In Section 4.3 (see also Section 6 at theend of the paper) we show that the particular case of affine mobi spaceis indeed the same as a module over a ring when the mobi algebra is aring.
Definition 3.1.
Let ( A, p, , ⁄ , be a mobi algebra. An A -mobi space ( X, q ) , consists of a set X and a map q : X × A × X → X such that: (X1) q ( x, , y ) = x (X2) q ( x, , y ) = y (X3) q ( x, a, x ) = x (X4) q ( x, ⁄ , y ) = q ( x, ⁄ , y ) = ⇒ y = y (X5) q ( q ( x, a, y ) , b, q ( x, c, y )) = q ( x, p ( a, b, c ) , y )The axioms (X1) to (X5) are the natural generalizations of axioms (A3) to (A7) of a mobi algebra. The natural generalization of (A8) would be q ( q ( x , a, y ) , ⁄ , q ( x , a, y )) = q ( q ( x , ⁄ , x ) , a, q ( y , ⁄ , y )) . (16)This condition, however, is too restrictive and is not in general verifiedby geodesic paths. That is the reason why we do not include it. When(16) is satisfied for all x , x , y , y ∈ X and a ∈ A , then we say thatthe A -mobi space ( X, q ) is affine and speak of an A -mobi affine space(see Section 4.3).Here are some immediate consequences of the axioms for a mobispace. Proposition 3.1.
Let ( A, p, , ⁄ , be a mobi algebra and ( X, q ) anA-mobi space. It follows that: (Y1) q ( y, a, x ) = q ( x, a, y ) (Y2) q ( y, ⁄ , x ) = q ( x, ⁄ , y ) (Y3) q ( x, a, q ( x, b, y )) = q ( x, a · b, y ) (Y4) q ( q ( x, a, y ) , b, y ) = q ( x, a ◦ b, y ) (Y5) q ( q ( x, a, y ) , ⁄ , q ( x, b, y )) = q ( x, a ⊕ b, y ) (Y6) q ( x, ⁄ , q ( x, a, y )) = q ( x, a, q ( x, ⁄ , y )) (Y7) q ( q ( x, a, y ) , ⁄ , q ( y, a, x )) = q ( x, ⁄ , y ) (Y8) q ( q ( q ( x, a, y ) , b, x ) , ⁄ , q ( x, b, q ( x, c, y )))= q ( x, ⁄ , q ( x, p ( a, b, c ) , y )) (Y9) q ( x, a, y ) = q ( y, a, x ) ⇒ q ( x, a, y ) = q ( x, ⁄ , y ) (Y10) q ( x, a, y ) = q ( x, b, y ) ⇒ q ( x, p ( a, t, b ) , y ) = q ( x, a, y ) , for all t .Proof. The following proof of (Y1) , bearing in mind (1), uses (X5) , (X2) and (X1) : q ( x, a, y ) = q ( x, p (1 , a, , y )= q ( q ( x, , y ) , a, q ( x, , y ))= q ( y, a, x ) . (Y2) follows directly from (A1) and (Y1) . Begining with (2), (Y3) is a consequence of (X5) and (X1) : q ( x, a · b, y ) = q ( x, p (0 , a, b ) , y )= q ( q ( x, , y ) , a, q ( x, b, y ))= q ( x, a, q ( x, b, y )) . Considering (12), property (Y4) follows from (Y1) and (Y3) . q ( q ( x, a, y ) , b, y ) = q ( y, b, q ( y, a, x ))= q ( y, b · a, x )= q ( y, a ◦ b, x )= q ( x, a ◦ b, y ) . J. P. FATELO AND N. MARTINS-FERREIRA
Considering (4), (Y5) is just a particular case of (X5) . To prove (Y6) ,we use (X5) , (6) and (X1) . q ( x, ⁄ , q ( x, a, y )) = q ( q ( x, , y ) , ⁄ , q ( x, a, y ))= q ( x, p (0 , ⁄ , a ) , y )= q ( x, p (0 , a, ⁄ ) , y )= q ( q ( x, , y ) , a, q ( x, ⁄ , y ))= q ( x, a, q ( x, ⁄ , y )) . The following proof of (Y7) is based on (Y1) , (X5) and (8); q ( q ( x, a, y ) , ⁄ , q ( y, a, x )) = q ( q ( x, a, y ) , ⁄ , q ( x, a, y ))= q ( x, p ( a, ⁄ , a ) , y )= q ( x, ⁄ , y ) . To prove (Y8) , we start with the important property (13) of the un-derlying mobi algebra and then get: q ( x, ⁄ · p ( a, b, c ) , y ) = q ( x, b · a ⊕ b · c, y ) ⇒ q ( x, p (0 , ⁄ , p ( a, b, c )) , y ) = q ( x, p ( b · a, ⁄ , b · c ) , y ) ⇒ q ( x, ⁄ , q ( x, p ( a, b, c ) , y )) = q ( q ( x, b · a, y ) , ⁄ , q ( x, b · c, y )) ⇒ q ( x, ⁄ , q ( x, p ( a, b, c ) , y ))= q ( q ( x, b, q ( x, a, y )) , ⁄ , q ( x, b, q ( x, c, y ))) . It is easy to see that ( Y9 ) is a direct consequence of ( Y7 ), while ( Y10 )is a consequence of (X3) and (X5) . (cid:3) The property ( Y9 ) shows that if someone is travelling from x to y and someone else is travelling from y to x (along a unique geodesicpath parametrized by q that connects x and y ) then they meet at q ( x, ⁄ , y ). The element ⁄ in a mobility algebra is understood as thepoint at which p ( a, ⁄ , b ) equals p ( b, ⁄ , a ). When interpreted in termsof a parameter on the parametrized operation q ( x, ⁄ , y ) on a mobilityspace it has the meaning that, although it may not be one half of thepath between x and y in the expected sense (see Examples 2 and 3 in[7]), it is nevertheless the position at half way in the sense of a metricin a metric space. Property ( Y10 ) shows that if a particle is at thesame place at two distinct moments a and b , then it stays there at anyinstant in-between.A related property, which is perhaps worthwhile studying, is thefollowing one:if there exist a = b with q ( x, a, y ) = q ( x, b, y ) then x = y. (17)This behaviour is desirable to model geodesics as shortest paths butit cannot be deduced from the axioms of a mobi space. Indeed, in Example 4.3, with h = −
1, we have q ( x, ( t, s ) , y ) = (1 − t + s ) x +( t − s ) y and hence, for every appropriate u , x and y , q ( x, ( t + u, s + u ) , y ) = q ( x, ( t, s ) , y ) . For example, q ( x, (1 , , y ) = q ( x, (1 / , − / , y ) = y . Nevertheless,this property is satisfied when the mobi algebra is the unit interval andthe mobi space is a space with unique geodesic paths. See the remarkfollowing Theorem 5.4. 4. Examples
In this section we give a list of examples and families of examples ofmobi spaces over an appropriate mobi algebra.4.1.
First examples.
In the list of examples below, the underlyingmobi algebra structure (
A, p, , ⁄ ,
1) is the closed unit interval, i.e. A = [0 , , , p ( a, b, c ) = (1 − b ) a + bc, for all a, b, c ∈ A. In each case, we present a set X and a ternary operation q ( x, t, y ) ∈ X ,for all x, y ∈ X , and t ∈ A , verifying the axioms of Definition 3.1. Example 4.1.
Mobi spaces ( X, q ) over the unit interval:(1) Euclidean spaces provide examples of the form X = R n ( n ∈ N ) and q ( x, t, y ) = (1 − t ) x + t y. (2) The positive real numbers with the metric inspired by the usualeuclidean distance after applying a transformation x exp( x ) provides an example of the form X = R + with q ( x, t, y ) = x − t y t . (3) Another example, still inspired by the euclidean metric but witha different transformation, is X = R + q ( x, t, y ) = xytx + (1 − t ) y . (4) The previous examples are all particular instances of a familyof examples which can be constructed as follows. Start with afunction F , invertible on a subset of the real numbers X ⊆ R ,and put q ( x, t, y ) = F − ((1 − t ) F ( x ) + t F ( y )) . For instance, Example 4.1(2) corresponds to F ( x ) = log x whileExample 4.1(3) to F ( x ) = x . Clearly, in Example 4.1(1), F ( x ) = x . Other type of examples.
The canonical formula for euclideanspaces, considered in Example 4.1(1) of the previous subsection, canbe adapted to provide examples of mobi spaces in subsets of R n . Example 4.2.
Further examples of mobi spaces ( X, q ) over the unitinterval.(1) The two subsets of the plane X =] − π, π ] × R or X = [0 , π [ × R with the formula q (( θ , z ) , t, ( θ , z )) = ((1 − t ) θ + t θ , (1 − t ) z + tz ) can be viewed as two different choices of geodesic paths on thecylinder { ( x, y, z ) ∈ R | x = cos θ, y = sin θ, ( θ, z ) ∈ X } , from the point (cos θ , sin θ , z ) to the point (cos θ , sin θ , z ) .For instance, if considering the point in the cylinder (cos( − π , sin( − π ,
0) = (cos( 7 π , sin( 7 π , and the point (cos( π , sin( π , , the two different parametrizations give two different paths be-tween them. Indeed, one path goes through (1 , , while theother goes thorough ( − , , . A third choice for a parametriza-tion, which corresponds to the shortest paths for any two pointson the cylinder, will be given in subsection 5.2.(2) Other examples are also possible, such as the set X = R + × R with the formula q (( x , x ) , t, ( y , y )) = ( x + ( y − x ) t, x + ( y − x ) y t (1 − t ) x + t y ) . (3) Or the set X = R + × R with the formula q (( x , x ) , t, ( y , y )) = ( x +( y − x ) t, x + y − x x + y (2 x t +( y − x ) t )) . (4) We can also consider the set X = R and the formula q (( x , x ) , t, ( y , y )) =( x + ( y − x ) t, x + ( y − x ) 3 x t + 3 x ( y − x ) t + ( y − x ) t x + x y + y ) , if ( x , y ) = (0 , , and q ((0 , x ) , t, (0 , y )) = (0 , x + t ( y − x )) . (5) The last three examples are just particular cases of the followingfamily of mobi spaces, where f is an injective real function ofone variable and X ⊆ R is any set for which the formula q (( x , x ) , t, ( y , y )) = (cid:18) x + ( y − x ) t, x + ( y − x ) f ( x + ( y − x ) t ) − f ( x ) f ( y ) − f ( x ) (cid:19) , if x = y , and q (( x, x ) , t, ( x, y )) = ( x, x + t ( y − x )) , defines a map q : X × [0 , × X → X . Examples 4.2(2), (3) and (4), are obtained as particular cases ofExample 4.2(5), respectively, with f ( x ) = x , f ( x ) = x and f ( x ) = x .So far we have considered examples of mobi spaces over the unitinterval. Here is an example with a different mobi algebra. Example 4.3.
For the mobi algebra ( A, p, , ⁄ , let us use A = (cid:8) ( t , t ) ∈ R : | t | ≤ t ≤ − | t | (cid:9) ⁄ = (cid:18) , (cid:19) ; 1 = (1 ,
0) ; 0 = (0 , p ( a, b, c ) = ( a − b a − b a + b c + b c ,a − b a − b a + b c + b c ) . And for the mobi space ( X, q ) , let us use X = [0 , and q ( x, ( t, s ) , y ) = (1 − t − h s ) x + ( t + h s ) y, with h = ± . Let us now turn to a special case of mobi spaces, namely affine mobispaces.
Affine mobi spaces.
Let (
A, p, , ⁄ ,
1) be a mobi algebra. Anaffine mobi space (over A ) is a mobi space ( X, q ) for which the condition q ( q ( x , a, y ) , ⁄ , q ( x , a, y )) = q ( q ( x , ⁄ , x ) , a, q ( y , ⁄ , y )) (18)is satisfied for every x , x , y , y ∈ X and a ∈ A .We observe that the (even) stronger condition q ( q ( x , a, y ) , b, q ( x , a, y )) = q ( q ( x , b, x ) , a, q ( y , b, y )) (19)for every x , x , y , y ∈ X and a, b ∈ A , is worthwhile studying due toits connection with Proposition 6.4 in [7].If ( X, q ) is an affine mobi space then we obtain a midpoint algebra( X, ⊕ ) by defining x ⊕ y = q ( x, ⁄ , y ) (see [5] for the particulars of thebinary operation ⊕ ). In a future work we will investigate the converse.That is, given a midpoint algebra ( X, ⊕ ), we will study under whichconditions it is obtained from an affine mobi space. Another interestingtopic is to study the collection of all mobi space structures which giverise to the same midpoint algebra, developing the concept of homologyfor mobi affine spaces.For the moment, let us mention that when we choose an origin e ∈ X in a given (non-empty) midpoint algebra and assuming that there isan abelian group structure associated to it (in the sense of [5]), then,the set of group endomorphisms on X has a ring structure and, by theuse of Proposition 2.2, we conclude that it gives rise to a mobi algebra.Recall that a midpoint algebra ( X, ⊕ ) with an associated abelian groupis a midpoint algebra with the property that for every x, y ∈ X thereis an element ( x + y ) ∈ X such that e ⊕ ( x + y ) = x ⊕ y , then itfollows that ( X, + , e ) is an abelian group with the property that themap x x + x is invertible, with inverse x e ⊕ x .This simple observation gives rise to an important characterization,to be further developed into a future work, which can be stated asfollows:The collection of mobi algebra homomorphisms from A to End ( X ) is in a one-to-one correspondence with thecollection of all affine mobi spaces ( X, q ) such that ⊕ isdetermined by q .It is remarkable how similar the notion of affine mobi space is fromthe notion of an affine vector space. In particular, when A has aninverse to ⁄ , say 2 ∈ A , then the operation + always exists and isexplicitly given by the formula x + y = q ( e, , q ( x, ⁄ , y )). In this case,we have precisely the notion of an A -module (see the last Section).Surprisingly, even when ( X, ⊕ ) does not allow an abelian group struc-ture, a similar characterization is still possible, at the expense of re-placing the ring End ( X ) with a more sophisticated structure, namely End ⊕ e ( X ) which is specially designed (Definition 2.3) to keep the anal-ogy with the previous result while extending it into a more generalsetting.The collection of mobi algebra homomorphisms from A to End ⊕ e ( X ) is in a one-to-one correspondence with thecollection of all affine mobi spaces ( X, q ) such that ⊕ isdetermined by q .This and other aspects of affine mobi spaces will be developed thor-oughly in the continuation of this study. However, in order to give aglimpse of the kind of results that are expected, we present an exampleof a mobi space constructed from a midpoint algebra. Example 4.4.
Consider the midpoint algebra ( X, ⊕ ) , with X = [0 , and x ⊕ y = x + y for every x, y ∈ X . The fixed element in X ischosen to be e = 0 . The mobi algebra is ( A, p, (0 , , ( , , (1 , with A = (cid:8) ( x, y ) ∈ R | | y | ≤ x ≤ − | y | (cid:9) and p ( a, b, c ) = ( a − b a − b a + b c + b c ,a − b a − b a + b c + b c ) . Let us now use Proposition 2.1 to generate an example of a mobispace from a homomorphism of mobi algebras. It is easily checked that,for any h ∈ [ − , , the map ϕ h, ( a ,a ) : A → End ⊕ e ( X ) , given by ϕ h, ( a ,a ) ( x ) = ( a + ha ) x is well defined in End ⊕ e ( X ) . Indeed, Definition 2.3 is verified with ϕ h, ( a ,a ) ( x ) = (1 − a − ha ) x and ˜ ϕ h, ( a ,a ) ( x, y ) = (1 − a − ha ) x + ( a + ha ) y. Note that ϕ h, ( a ,a ) = ϕ h, ( a ,a because ( a , a ) = (1 − a , − a ) followsfrom (1). However, ϕ h, ( a ,a ) is an homomorphism of mobi algebras ifand only if h = ± . In these cases, the mobi operations on X , definedby ⊕ q ( x, ( t, s ) , y ) = ϕ ± , ( t,s ) ( x ) ⊕ ϕ ± , ( t,s ) ( y ) , are given by: q ( x, ( t, s ) , y ) = (1 − t ∓ s ) x + ( t ± s ) y. The resulting mobi space (
X, q ) is the same as the one in Exam-ple 4.3.We will now see some examples obtained from physics.
Examples with physical interpretation.
This section endswith Example 4.5, obtained from the motion of a projectile, some com-ments on counter-examples, as well as a general example from mechan-ics.
Example 4.5.
For any k ∈ R , we may form a mobi space ( X, q ) overthe unit interval by taking the set X = R with the formula q (( x , x ) , t, ( y , y )) =((1 − t ) x + t y + k ( y − x ) (1 − t ) t, (1 − t ) x + t y ) . In this example, when y > y , the operation q (( x , y ) , t, ( x , y )) sim-ply gives the position at instant t of a projectile in one-dimensionalclassical mechanics with constant acceleration a x = − k , that is mov-ing from position x at time y to position x at time y . Example 4.6.
If a one-dimensional space X would have been consid-ered to describe the motion of the projectile instead of the Euclideanspace-time of Example 4.5, the ternary operation would have been: q ( x , t, x ) = (1 − t ) x + t x + k (1 − (1 − t ) t. This operation does not verify all the axioms of Definition 3.1. In par-ticular, if the particle is not at rest, we will never get the idempotency q ( x, t, x ) = x . Axiom (X5) is not verified either. Most of the operations q : X × A × X → X that we can think ofwill not verify some of the axioms of Definition 3.1. Let us just pointout that a simple example such as q ( x, t, y ) = x + t ( y − x ), for x, y insome subset of R n and t in a non trivial subset of R , does not in generalverify (X5) .There are also examples where an operation q might verify the axiomsof a mobi space but has no mobi space associated (simply because itmay not be everywhere defined). That would be the case if Example 4.5would be generalized to Special Relativity [4]. However, in Minkowskispace-time, not every two points can be reached from one another ifone point is not inside the light cone of the other. Example 4.7.
Finally, let us observe that, in general, the solutionsof ¨ x = f ( x, ˙ x, t ) (where ˙ x is the derivative of x with respect to t ),in R n , will not verify the axioms of a mobi space. Nevertheless, in R n +1 , where the extra dimension is time, and if every two points canbe reached from one another, then we can construct a mobi space asexplain in Corollary 5.6. The case f ( x, ˙ x, t ) = − k gives Example 4.5.To show another example, let us look at a critically damped harmonicoscillator corresponding to f ( x, ˙ x, t ) = − k x − k ˙ x . For any k ∈ R , we obtain the following mobi space ( R , q ) over the unit interval with theformula q (( x , x ) , t, ( y , y )) =((1 − t ) x e kt ( x − y ) + t y e k (1 − t )( y − x ) , (1 − t ) x + t y ) . In the following section we will thoroughly analyse examples occur-ring from spaces with geodesic paths.5.
Examples from Geodesics
In this section we analyse spaces which satisfy the equation forgeodesics and observe that they all give rise to a mobi space. Wehave decided, for simplicity, to express the results within the scope of R n , but it is clear that the same principle will be valid for pseudo-Riemmanian manifolds with appropriate tensor metrics. This would,however, require a more sophisticated level of abstraction which is be-yond our current purpose, namely to show the existence of examplesof mobi spaces that are obtained from spaces with geodesics.We also show that under suitable choices for appropriate sections,every identification space inherits the mobi space structure from itscovering space. We illustrate this concept with the case of the cylinder(see Example 4.2(1) and Example 5.2).5.1. Spaces with unique geodesics.
Let
X, V ⊆ R n be two opensubsets of the n -dimensional space, with V a vector space, and g : X × V → R n a map such that g ( x, λv ) = λ g ( x, v ) , (20)for all x ∈ X , v ∈ V and λ ∈ R . Moreover, the initial value problem (cid:26) x ′ = v , x (0) = x ∈ Xv ′ = g ( x, v ) , v (0) = v ∈ V (21)is supposed to have a unique solution, denoted by x ( t ) = π ( x , v , t ) , for every pair ( x , v ) ∈ X × V of initial conditions. This means that,for fixed x ∈ X and v ∈ V , given any continuous map f : R → X , withcontinuous derivatives f ′ : R → V and f ′′ : R → R n , if f (0) = x, f ′ (0) = v, f ′′ ( t ) = g ( f ( t ) , f ′ ( t )) (22)for every t ∈ R , then f ( t ) = π ( x, v, t ) . (23) See Example 5.1 for an illustration. Of course, the function π has thefollowing properties: π ( x, v,
0) = x (24) π ′ ( x, v,
0) = v (25) π ′′ ( x, v, s ) = g ( π ( x, v, s ) , π ′ ( x, v, s )) , (26)for every s ∈ R , where π ′ ( x, v, s ) stands for the derivative with respectto the variable s . Lemma 5.1.
Consider an initial value problem such as (21) . Thefollowing conditions hold for every x ∈ X , v ∈ V , and s, t, u ∈ R : π ( x, , t ) = x (27) π ( x, v, s + u t ) = π ( π ( x, v, s ) , u π ′ ( x, v, s ) , t ) . (28) Proof.
Considering the map f ( t ) = x for the first case and the map f ( t ) = π ( x, v, s + ut ) for the second case, f satisfies the conditions (22)for each case. The desired equalities then follow from the uniquenessof solutions with respect to initial conditions, as expressed in condi-tion (23). (cid:3) Some useful particular cases of the previous lemma are: π ( x, v, s t ) = π ( x, s v, t ) (29) π ( x, v, − t ) = π ( π ( x, v, , − π ′ ( x, v, , t ) (30) π ( x, v, t ) = π ( π ( x, v, s ) , ( t − s ) π ′ ( x, v, s ) , . (31)Property (31) follows from π ( x, v, t ) = π ( x, v, s + ( t − s ) 1). Lemma 5.2.
Consider an initial value problem such as (21) and sup-pose the existence of a map β : X × X → V such that π ( x, β ( x, y ) ,
1) = y, (32) for every x, y ∈ X . Then the following two conditions are equivalent:(a) π ( x, v ,
1) = π ( x, v , ⇒ v = v , ∀ x ∈ X, v , v ∈ V (b) the map β with the property (32) is unique.Moreover, we always have: β ( x, y ) = β ( x, y ) ⇒ y = y . Proof.
Assuming condition ( a ), if there exists another map with thesame property as β , say ˆ β : X × X → V , then, for every x, y ∈ X , wehave π ( x, β ( x, y ) ,
1) = y = π ( x, ˆ β ( x, y ) , . Now, from ( a ) it immediately follows that β = ˆ β . Conversely, let begiven any x ∈ X and v , v ∈ V such that π ( x, v ,
1) = π ( x, v , and let us denote by y ∈ X , both π ( x, v ,
1) and π ( x, v , β ( x, y ) ∈ V and observe that also π ( x, β ( x, y ) ,
1) = y andhence, by the uniqueness of β , we obtain β ( x, y ) = v = v , which concludes the first part of the proof. For the remaining part wesimply observe that if β ( x, y ) = β ( x, y ) then y = π ( x, β ( x, y ) ,
1) = π ( x, β ( x, y ) ,
1) = y . (cid:3) An interesting consequence of Lemma 5.2, when applied togetherwith property (29), is that within an initial value problem such as (21)where there exists a unique β verifying (32), we have, for s = 0, x ∈ X and v , v ∈ V , π ( x, v , s ) = π ( x, v , s ) ⇒ π ( x, s v ,
1) = π ( x, s v , ⇒ v = v . (33) Lemma 5.3.
Consider an initial value problem such as (21) . If thereexists a unique β : X × X → V such that, for all x, y ∈ X , π ( x, β ( x, y ) ,
1) = y (34) then we also have: β ( x, x ) = 0 (35) s β ( x, y ) = β ( x, π ( x, β ( x, y ) , s )) (36) β ( π ( x, v, s ) , π ( x, v, t )) = ( t − s ) π ′ ( x, v, s ) . (37) for every x ∈ X , v ∈ V and t, s ∈ R .Proof. From (27), we have π ( x, ,
1) = x and so β ( x, x ) = 0 because β ( x, x ) is the unique element in V with the property π ( x, β ( x, x ) ,
1) = x. Similarly we conclude, from (29) with t = 1, that (36) holds, indeed β ( x, π ( x, β ( x, y ) , s )) is the unique element in V such that π ( x, β ( x, π ( x, β ( x, y ) , s )) ,
1) = π ( x, β ( x, y ) , s ) . Finally, we observe that from (34) we get: π ( π ( x, v, s ) , β ( π ( x, v, s ) , π ( x, v, t )) ,
1) = π ( x, v, t )which, using (31), implies π ( π ( x, v, s ) , ( t − s ) π ′ ( x, v, s ) , π ( π ( x, v, s ) , β ( π ( x, v, s ) , π ( x, v, t )) , . Therefore, the unicity of β proves (37). (cid:3) A particular case of the previous lemma is: π ′ ( x, β ( x, y ) ,
1) = − β ( y, x ) . (38) Theorem 5.4.
Consider an initial value problem such as (21) in whichthere is a unique β : X × X → V such that (34) holds. Then, the struc-ture ( X, q ) , with q ( x, t, y ) = π ( x, β ( x, y ) , t ) is a mobi space over the mobi algebra ([0 , , p, , , , where p ( s, u, t ) = s + u ( t − s ) . Proof. (X1) is simply a consequence of (24) and (X2) of (34): q ( x, , y ) = π ( x, β ( x, y ) ,
0) = x,q ( x, , y ) = π ( x, β ( x, y ) ,
1) = y. To prove (X3) , we use (35) and (27): q ( x, t, x ) = π ( x, β ( x, x ) , t ) = π ( x, , t ) = x. Lemma 5.2 and property (33) imply (X4) : q ( x, ⁄ , y ) = q ( x, ⁄ , y ) = ⇒ π ( x, β ( x, y ) , ⁄ ) = π ( x, β ( x, y ) , ⁄ )= ⇒ β ( x, y ) = β ( x, y )= ⇒ y = y . Using (28) and (37), we obtain (X5) : q ( q ( x, s, y ) , u, q ( x, t, y ))= π ( π ( x, β ( x, y ) , s ) , β ( π ( x, β ( x, y ) , s ) , π ( x, β ( x, y ) , t )) , u )= π ( π ( x, β ( x, y ) , s ) , ( t − s ) π ′ ( x, β ( x, y ) , s ) , u )= π ( x, β ( x, y ) , s + u ( t − s ))= q ( x, p ( s, u, t ) , y ) . (cid:3) Note that, when s = 0, from (33), we also deduce that q ( x, s, y ) = q ( x, s, y ) implies y = y .Furthermore, in this case, the property (17) is verified. Corollary 5.5. If S ⊆ R n is a Riemann surface with a unique geodesicpath between any two points, then ( S, q ) is a mobi space with q ( x, t, y ) being the position at an instant t on the geodesic path between x and y .Proof. If Γ kij : S → R are the Christoffel symbols (see [11], for instance)for the metric, then the function g in the initial value problem (21) isof the form g k ( x, y ) = − X i,j y i Γ kij ( x ) y j and henceforth satisfies (20). The existence of unique geodesic pathsbetween any two points implies the uniqueness of β in Theorem 5.4. (cid:3) Corollary 5.6.
Consider the functions f : R n × R n × R → R n and x : R → R n , and let ˙ x and ¨ x be the first and second derivatives of x with respect to the variable t ∈ R . If the following problem ¨ x = f ( x, ˙ x, t ) x ( t ) = x x ( t ) = x (39) has a unique solution for any x , x ∈ R n and t , t ∈ R , t = t ,expressed as x ( t ) = F ( x , t , x , t , t ) , then ( R n +1 , q ) is a mobi spaceover the mobi algebra ([0 , , p, , , , with p ( t , s, t ) = t + s ( t − t ) ,q (( x , t ) , s, ( x , t )) = ( F ( x , t , x , t , p ( t , s, t )) , p ( t , s, t )) . Proof.
Considering the variable s such that p ( t , s, t ) = t , and denotingby x ′ and x ′′ the first and second derivatives of x with respect to s, weget x ′ = t ′ ˙ x and x ′′ = t ′ ¨ x . With χ = ( x, t ) ∈ R n +1 and ω = ( x ′ , t ′ ), wethen have: (cid:26) χ ′ = ωω ′ = g ( χ, ω ) , with g (( x, t ) , ( x ′ , t ′ )) = ( t ′ f ( x, x ′ t ′ , t ) , . It is obvious that g ( χ, λω ) = λ g ( χ, ω ). The unicity of a solution of(39) implies that there exists a initial value problem (at the initial value s = 0), such as (21), equivalent to (39). Therefore, Theorem 5.4 canbe applied and proves this Corollary. (cid:3) The mobi spaces of Examples 4.5 and 4.7 are illustrations of theresult of Corollary 5.6.
Example 5.1.
To illustrate the result of Theorem 5.4, consider theexplicit example where X = R , V = R and, with k ∈ R , g : X × V → R (( a , a ) , ( w , w )) → ( − kw , . The solution of the following initial value problem, involving the func-tions x : R → X and v : R → V , (cid:26) x ′ = v , x (0) = ( x , x ) ∈ Xv ′ = g ( x, v ) , v (0) = ( v , v ) ∈ V (40) is π (( x , x ) , ( v , v ) , t ) = ( x + v t − kv t , x + v t ) It is easy to see that there is a unique β : X × X → V such that π (( x , x ) , β (( x , x ) , ( y , y )) ,
1) = ( y , y ) given by: β (( x , x ) , ( y , y )) = ( y − x + k ( y − x ) , y − x ) . We then obtain the ternary operation q : X × [0 , × X → X of themobi space ( X, q ) which is given by: q (( x , x ) , t, ( y , y )) = π (( x , x ) , β (( x , x ) , ( y , y )) , t )= ( x + ( y − x ) t + k ( y − x ) ( t − t ) , x + ( y − x ) t ) . This is, in fact, Example 4.5. For an explicit example of geodesics ona two dimensional surface, we refer to [6] . Metric spaces in which there is a unique geodesic through any twopoints have been characterized in [1] (see also [8]).5.2.
The cylinder as an identification space and its geodesics.
If the uniqueness of the geodesic paths in a Riemannian surface is notguaranteed, we may still choose appropriate values for the map β as inProposition 5.4 and obtain a mobi space. For example, although, ona cylinder, there are infinitely many geodesics through two points thatdo not lie on the same circle, we can construct mobi spaces over theset of points of a cylinder as illustrated in Example 4.2(1) of Section 4.See, for example [2] for the study of spaces in which a geodesic throughtwo points is not unique. Proposition 5.7.
Let ( X, q ) be a mobi space over the mobi algebra ( A, p, , ⁄ , and h : X → S a map onto a set S . If we can find twomaps s : S → X and θ : S × S → X , satisfying the following conditionsfor every u, v, v ′ ∈ S a , a , a ∈ Ahs ( u ) = uhθ ( u, v ) = vθ ( u, u ) = s ( u ) hq ( s ( u ) , ⁄ , θ ( u, v )) = hq ( s ( u ) , ⁄ , θ ( u, v ′ )) ⇒ v = v ′ hq ( s ( u ) , p ( a , a , a ) , θ ( u, v )) = hq ( shq ( s ( u ) , a , θ ( u, v )) , a , · · · θ ( hq ( s ( u ) , a , θ ( u, v )) , hq ( s ( u ) , · · · a , θ ( u, v )))) then ( S, q S ) is a mobi space over A with q S ( u, a, v ) = hq ( s ( u ) , a, θ ( u, v )) , for every u, v ∈ S and a ∈ A .Proof. It is clear that the axioms of a mobi space follow from the con-ditions that are assumed to be satisfied by the three maps. (cid:3)
As an example we show how to obtain geodesic paths on a cylinderfrom geodesic paths on a plane.
Example 5.2. let ( X, q ) be the euclidean plane with the usual geodesicpaths over the mobi algebra of the unit interval. Denote by S the set [0 , π [ × R and define three maps: h , s and θ . The map h : X → S is defined as h ( x, y ) = ( x mod 2 π, y ) , the map s : S → X is the inclusionmap and θ : S × S → X is defined as θ (( x , y ) , ( x , y )) = ( x + 2 π, y ) if x − x < − π ( x , y ) if − π ≤ x − x ≤ π ( x − π, y ) if π < x − x . It can be checked that these maps satisfy the conditions of the previousproposition and hence they define a mobi space on the set S . The resulting mobi space structure in this case is not the same asthe ones presented in Example 4.2(1). Indeed, in this case, we obtaina parametrization for the cylinder which induces, via the mobi space,the shortest path between any two points. That was not the case inExample 4.2(1). 6.
Comparison with R-modules
Consider a unitary ring ( R, + , · , , R contains the inverse of 1 + 1, then it is a mobi algebra and if amobi algebra ( A, p, , ⁄ ,
1) contains the inverse of ⁄ , in the sense of(2), then it is a unitary ring. In this section, we will compare a moduleover a ring R with a mobi space over a mobi algebra A. First, let usjust recall that a module over a ring R is a system ( M, + , e, ϕ ), where ϕ : R → End ( M ) is a map from R to the usual ring of endomorphisms,such that ( M, + , e ) is an abelian group and ϕ is a ring homomorphism.The following Theorem shows how to construct a mobi space from amodule over a ring containing the inverse of 2. Theorem 6.1.
Consider a module ( X, + , e, ϕ ) over a unitary ring ( A, + , · , , . If A contains (1 + 1) − = ⁄ then ( X, q ) is an affinemobi space over the mobi algebra ( A, p, , ⁄ , , with p ( a, b, c ) = a + bc − ba (41) q ( x, a, y ) = ϕ − a ( x ) + ϕ a ( y ) . (42) Proof. ( A, p, , ⁄ ,
1) is a mobi algebra by Theorem 7.2 of [7]. We showhere that the axioms of Definition 3.1, as well as (16), are verified. Thefirst three axioms are easily proved: q ( x, , y ) = ϕ ( x ) + ϕ ( y ) = x + e = xq ( x, , y ) = ϕ ( x ) + ϕ ( y ) = e + y = yq ( x, a, x ) = ϕ − a ( x ) + ϕ a ( x ) = ϕ − a + a ( x ) = ϕ ( x ) = x. Axiom (X4) is due to the fact that ⁄ + ⁄ = 1 and consequently ϕ ⁄ ( y ) = ϕ ⁄ ( y ) ⇒ ϕ ⁄ ( y ) + ϕ ⁄ ( y ) = ϕ ⁄ ( y ) + ϕ ⁄ ( y ) ⇒ ϕ ( y ) = ϕ ( y ) ⇒ y = y . Next, we give a proof of Axiom (X5) . It is relevant to notice that,besides other evident properties of the module X , the associativityof + plays an important part in the proof: q ( q ( x, a, y ) , b, q ( x, c, y ))= ϕ − b ( ϕ − a ( x ) + ϕ a ( y )) + ϕ b ( ϕ − c ( x ) + ϕ c ( y ))= ϕ − b ( ϕ − a ( x )) + ϕ − b ( ϕ a ( y )) + ϕ b ( ϕ − c ( x )) + ϕ b ( ϕ c ( y ))= ϕ (1 − b )(1 − a ) ( x ) + ϕ b (1 − c ) ( x ) + ϕ (1 − b ) a ( y ) + ϕ bc ( y )= ϕ − a + ba − bc ( x ) + ϕ a − ba + bc ( y )= ϕ − p ( a,b,c ) ( x ) + ϕ p ( a,b,c ) ( y )= q ( x, p ( a, b, c ) , y ) . It remains to prove (16): q ( q ( x , a, y ) , ⁄ , q ( x , a, y ))= ϕ ⁄ ( ϕ − a ( x ) + ϕ a ( y )) + ϕ ⁄ ( ϕ − a ( x ) + ϕ a ( y ))= ϕ ⁄ ( ϕ − a ( x ) + ϕ a ( y ) + ϕ (1 − a ) ( x ) + ϕ a ( y ))= ϕ ⁄ ( ϕ − a ( x + x ) + ϕ a ( y + y ))= ϕ (1 − a ) ⁄ ( x + x ) + ϕ a ⁄ ( y + y )= ϕ (1 − a ) ( ϕ ⁄ ( x ) + ϕ ⁄ ( x )) + ϕ a ( ϕ ⁄ ( y ) + ϕ ⁄ ( y ))= ϕ (1 − a ) ( q ( x , ⁄ , x )) + ϕ a ( q ( y , ⁄ , y ))= q ( q ( x , ⁄ , x )) , a, q ( y , ⁄ , y )) . (cid:3) Theorem 6.2.
Consider a mobi space ( X, q ) , with a fixed chosen ele-ment e ∈ X , over a mobi algebra ( A, p, , ⁄ , . If A contains suchthat p (0 , ⁄ ,
2) = 1 then ( X, + , e, ϕ ) is a module over the unitary ring ( A, + , · , , , with a + b = p (0 , , p ( a, ⁄ , b )) (43) a · b = p (0 , a, b ) (44) ϕ a ( x ) = q ( e, a, x ) (45) x + y = q ( e, , q ( x, ⁄ , y )) = ϕ ( q ( x, ⁄ , y )) . (46) Proof. ( A, + , · , ,
1) is a unitary ring by Theorem 7.1 of [7]. We provehere that ( X, + , e, ϕ ) is a module over A . First, we observe that, usingin particular (Y3) of Proposition 3.1, we have: q ( e, ⁄ , x + y ) = q ( e, ⁄ , q ( e, , q ( x, ⁄ , y )))= q ( e, ⁄ · , q ( x, ⁄ , y ))= q ( e, , q ( x, ⁄ , y ))= q ( x, ⁄ , y ) . Then, the property (16) of an affine mobi space is essential to provethe associativity of the operation + of the module: q ( e, ⁄ , q ( e, ⁄ , ( x + y ) + z )) = q ( q ( e, ⁄ , e ) , ⁄ , q ( x + y, ⁄ , z ))= q ( q ( e, ⁄ , x + y ) , ⁄ , q ( e, ⁄ , z ))= q ( q ( x, ⁄ , y ) , ⁄ , q ( e, ⁄ , z ))= q ( q ( x, ⁄ , e ) , ⁄ , q ( y, ⁄ , z ))= q ( q ( e, ⁄ , x ) , ⁄ , q ( e, ⁄ , y + z ))= q ( q ( e, ⁄ , e ) , ⁄ , q ( x, ⁄ , y + z ))= q ( e, ⁄ , q ( e, ⁄ , x + ( y + z )))Which, by (X4) , implies that ( x + y ) + z = x + ( y + z ). Commutativityof + and the identity nature of e are easily proved: q ( e, ⁄ , e + x ) = q ( e, ⁄ , x ) ⇒ e + x = xq ( e, ⁄ , x + y ) = q ( x, ⁄ , y ) = q ( y, ⁄ , x )= q ( e, ⁄ , y + x ) ⇒ x + y = y + x. Cancellation is achieved with − x = q ( e, p (1 , , , x ). Indeed: q ( e, ⁄ , q ( e, p (1 , , , x ) + x ) = q ( q ( e, p (1 , , , x ) , ⁄ , x )= q ( q ( e, p (1 , , , x ) , ⁄ , q ( e, , x ))= q ( e, p ( p (1 , , , ⁄ , , x )= q ( e, p (1 , p (2 , ⁄ , , , x )= q ( e, p (1 , , , x )= q ( e, , x ) = e = q ( e, ⁄ , e )To prove that ϕ a ( x + y ) = ϕ a ( x ) + ϕ a ( y ), we will again need (16): q ( e, ⁄ , ϕ a ( x + y )) = q ( e, ⁄ , q ( e, a, x + y ))= q ( q ( e, a, e ) , ⁄ , q ( e, a, x + y ))= q ( q ( e, ⁄ , e ) , a, q ( e, ⁄ , x + y ))= q ( e, a, q ( x, ⁄ , y ))= q ( q ( e, a, x ) , ⁄ , q ( e, a, y ))= q ( ϕ a ( x ) , ⁄ , ϕ a ( y ))= q ( e, ⁄ , ϕ a ( x ) + ϕ a ( y )) . To prove that ϕ a + b ( x ) = ϕ a ( x )+ ϕ b ( x ), let us first recall that, in a mobialgebra with 2 and a + b = p (0 , , p ( a, ⁄ , b )), we have the followingproperty: p (0 , ⁄ , a + b ) = p ( a, ⁄ , b ) . We then have q ( e, ⁄ , ϕ a + b ( x )) = q ( e, ⁄ , q ( e, a + b, x ))= q ( q ( e, , x ) , ⁄ , q ( e, a + b, x ))= q ( e, p (0 , ⁄ , a + b ) , x )= q ( e, p ( a, ⁄ , b ) , x )= q ( q ( e, a, x ) , ⁄ , q ( e, b, x ))= q ( ϕ a ( x ) , ⁄ , ϕ b ( x ))= q ( e, ⁄ , ϕ a ( x ) + ϕ b ( x )) . Last two properties are easily proved: ϕ a · b ( x ) = q ( e, a · b, x ) = q ( e, a, q ( e, b, x )) = ϕ a ( ϕ b ( x )) ϕ ( x ) = q ( e, , x ) = x. (cid:3) Proposition 6.3.
Consider a R-module ( X, + , e, ϕ ) within the condi-tions of Theorem 6.1 and the corresponding mobi space ( X, q ) . Thenthe R-module obtained from ( X, q ) by Theorem 6.2 is the same as ( X, + , e, ϕ ) .Proof. From (
X, q ), we define x + ′ y = q ( e, , q ( x, ⁄ , y )) and ϕ ′ ( x ) = q ( e, a, x )and obtain the following equalities: x + ′ y = e + ϕ ( q ( x, ⁄ , y ))= ϕ ( ϕ ⁄ ( x ) + ϕ ⁄ ( y ))= ϕ · ⁄ ( x ) + ϕ · ⁄ ( y )= x + yϕ ′ a ( x ) = ϕ − a ( e ) + ϕ a ( x ) = e + ϕ a ( x ) = ϕ a ( x ) . (cid:3) Proposition 6.4.
Consider an affine mobi space ( X, q ) within theconditions of Theorem 6.2 and the corresponding module ( X, + , e, ϕ ) .Then the affine mobi space obtained from ( X, + , e, ϕ ) by Theorem 6.1is the same as ( X, q ) .Proof. From ( X, + , e, ϕ ), we define q ′ ( x, a, y ) = ϕ − a ( x ) + ϕ a ( y )and obtain the following equalities: q ′ ( x, a, y ) = q ( e, a, x ) + q ( e, a, y )= q ( x, a, e ) + q ( e, a, y )= q ( e, , q ( q ( x, a, e ) , ⁄ , q ( e, a, y ))) . Now, because we are considering that (
X, q ) is affine, we get: q ′ ( x, a, y ) = q ( q ( e, a, e ) , , q ( q ( x, ⁄ , e ) , a, q ( e, ⁄ , y )))= q ( q ( e, , q ( e, ⁄ , x )) , a, q ( e, , q ( e, ⁄ , y )))= q ( q ( e, · ⁄ , x ) , a, q ( e, · ⁄ , y ))= q ( x, a, y ) . (cid:3) We have completely characterized affine mobi spaces in terms ofmodules over a unitary ring in which 2 is invertible. In a sequel tothis paper we will investigate how to characterize mobi spaces in termsof homomorphisms between mobi algebras. We will also dedicate ourattention to the case of non affine mobi spaces.We finish this section by taking a closer look to Example 4.5 andExample 4.2(4).Example 4.5 is an example of an affine mobi space and can be ex-tended to the case where the underlying mobi algebra is ( R , p, , , R , + , · , ,
1) given by( R , + , (0 , , ϕ ) with( x , x ) + ( y , y ) = ( x + y − kx y , x + y ) ϕ a ( x , x ) = ( ax + k (1 − a ) ax , ax ) . By Theorem 6.1, we can construct a mobi space from this module andverify that it is the same as Example 4.5. Of course, in this example,the module is a vector field and there is a homomorphism, namely f ( x , x ) = ( x + k ( x − x ) , x )from ( R , + , (0 , , ϕ ) to the usual vector fied in R .Example 4.2(4) is a mobi space that it is not affine. Indeed (16) isnot always verified as, for instance, we have: q ( q ((0 , , , (1 , , , q ((1 , , , (0 , , q ( q ((0 , , , (1 , , , q ((1 , , , (0 , ,
112 ) . We can also extend its underlying mobi algebra to ( R , p, , , ϕ a functions. Choosing e = (0 , x , x ) + ( y , y )= (cid:18) x + y , ( x + 4 x y + 7 y ) x + (7 x + 4 x y + y ) y x + x y + y (cid:19) for ( x , y ) = (0 ,
0) and(0 , x ) + (0 , y ) = (0 , x + 4 y ); ϕ a ( x , x ) = ( ax , a x ) . The operation + is commutative, admits e as a identity and the sym-metric element ( − x , − x ) for all ( x , x ) ∈ R but it is not associative.On the other side, we have: ϕ a ( x + y ) = ϕ a ( x ) + ϕ a ( y ) ϕ a + b ( x ) = ϕ a ( x ) + ϕ b ( x ) ϕ ab ( x ) = ϕ a ( ϕ b ( x )) ϕ ( x ) = xϕ ( x ) = e Beginning with the structure ( X, + , e, ϕ ), we can construct a ternaryoperation q on X given by (42) but then ( X, q ) is not a mobi spacebecause it doesn’t verify axiom (X5) . This example shows that a mobispace is richer than a structure of the type ( X, + , ϕ ). References [1] H. Busemann,
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