On the geometric concomitants

Abstract In this note the necessary and sufficient condition it would the concomitant of the geometric object was the geometric object too is given.


Introduction
S. Gołąb introduced the notion of the concomitant of an object in [4]. Let S be the set of elements called the objects and let T be the set of elements called the transformations, for which is defined the binary operation • : T × T → T the superposition of these transformations. The objects in S are said to be geometric if there exists a function F : S × T → S (called the transformation law) which transforms an object s ∈ S by a transformation t ∈ T to the object F (s, t) and such that the diagram s F (s, t 1 ) [54]

Zenon Moszner
Let ϕ be a function from the set S to the set S 1 of objects. The object ϕ(s) for s ∈ S is said to be the concomitant of the object s. For the geometric object s with the transformation law F its concomitant ϕ is said to be geometric if there exists a transformation law G : (the concomitant equation). If objects in S are geometric their concomitants do not have to be the geometric objects. Hence the general question can be stated: when the concomitant of the geometric object is a geometric object too? We have not found the answer to this question in the rich literature about the concomitants (see, e.g. the bibliography in [1] and [5]).

Main considerations
if and only if (ii) The functions of the form where f : Proof. First we prove (i) and (ii). If ϕ(x) = ϕ(y), then The function G defined in (3) is well defined by (2). The relation (1) To show (iii) notice that if the concomitant ϕ is geometric, then there exists a solution G : Interpretation of the implication (2). The condition (2) means that the family of the levels of the function ϕ, i.e. the family {ϕ −1 (a) : a ∈ S 1 }, is such that the image of the level by the function from the family {F (., t) : t ∈ G} is included in some level.

Examples
Example 3.1 An injection and a constant function are evidently the solutions of the conditional equation (2) for every function F . If ϕ is an injection, then G( where h is a bijection of R and a ∈ R, is a solution of (2) for the function F ( The function

Zenon Moszner
is the solution of (2) for the function F (x, t) = x exp t. We have G = F in this case.
Example 3.4 Let (G, +) be a group and let F ( } x∈G is the family of the right cosets of the group G for a subgroup G * and Φ is an injection from F to G. If ϕ is of this form, then (2) has the form .
Assume now that ϕ is a solution of (2). We define a relation R on G as follows

It is an equivalence relation and xRy implies (x + t)R(y + t).
From here the family of the equivalence classes of R is the same as the family F of the right cosets of the group G for a subgroup [3]. Let S be a selector of this family.
Example 3.5 The injection from R to R and a constant function are the only solutions of (2) for the geometric object x = v2 v1 , where v 1 , v 2 are the coordinates of the 2-dimensional contravariant vector. Indeed, this object has the transformation law of the form [1]), thus (2) is of the form For A 1 2 = A 2 1 = 1 and A 2 2 = 0 we have From here the family F of the levels of ϕ is invariant by the arbitrary translations. This yields that this family F = R/R * , where R/R * is the quotient group of R for some subgroup R * (see the previous example). Assume that {0} = R * = R. This implies that there exist x 1 , x 2 such that x 1 = 0 and ϕ(0) = ϕ(x 1 ) = ϕ(x 2 ). We have by (4) that for every x, y, a ∈ R. From here thus the contradiction. If R * = {0}, then ϕ is an injection, if R * = R, then ϕ is a constant function.
On the geometric concomitants

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Example 3.6 Let the solution of translation equation F : R × R → R be of the form where I n for n ∈ K 1 ⊂ N are open and disjoint non-empty intervals and h n : I n → R are bijections. Put K 2 = R \ I n and K = K 1 ∪ K 2 . The function ϕ of the form where α : K → K is such that α(K 2 ) ⊂ K 2 and β is an arbitrary function from K 1 to R, is the solution of (2), since ϕ[F (x, t)] = F (ϕ(x), t). Every solution ϕ of the last equation is of the form (6) (see [6]). If K 1 = ∅, then the function ϕ(x) = x 0 ∈ I k for some k ∈ K 1 and x ∈ R is the solution of (2) and it is not the solution of the equation for the continuous function F . Indeed, if the constant function ϕ(x) = c is a solution of (7), then F (c, t) = c for t ∈ G and x 0 / ∈ R \ I n . This yields that the implication (2) and the equation (7) are not equivalent.
The solution F : R × R → R of the translation equation is of the form (5) if it is continuous with respect to the second variable for every x ∈ S and for which at least one of the functions F (., t) : R → R is continuous and F (x, 0) = x for x ∈ R ( [7], this form of F is proved in [8] if it is continuous). Notice that F is of this form too if it is Carathéodory, i.e. the function F (x, .) : R → R is measurable for every x ∈ R and the function F (., t) : R → R is continuous for every t ∈ R, since it is continuous in this case (see [2]).

Remarks
Remark 4.1 The operation "+" in G occurs not explicitly in the implication (2), it is "hidden" in the function F . From here the solution of (2) depends on "+" (see the above   (1) is not unique since we have in this case many idempotents from S 1 \ ϕ(S) to S 1 \ ϕ(S).

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Zenon Moszner be a solutions of the translation equation. Moreover, let λ : G 1 → G 2 be a homomorphism and let ϕ be the function from S 1 to S 2 . The generalized concomitant equation in which F and G are the given solutions of the translation equation, is solved in [6] (notice that the typing errors are in this paper!). In [6], among others, it is proved that the multiplication by scalar is the only geometric concomitant of the vector.
Remark 4.4 The equation (8) occurs in the theory of abstract automata in the concept of their homomorphism [9].