# Trace- and pseudo-products: Restriction-like semigroups with a band of
projections

D. G. FitzGerald & M. K. Kinyon  School of Natural Sciences, University of
Tasmania, POB 37 Hobart, Australia 7001  Department of Mathematics, University
of Denver, Denver CO 80208

###### Abstract.

We ascertain conditions and structures on categories and semigroups which
admit the construction of pseudo-products and trace products respectively,
making their connection as precise as possible. This topic is modelled on the
ESN Theorem and its generalisation to ample semigroups. Unlike some other
variants of ESN, it is self-dual (two-sided), and the condition of commuting
projections is relaxed. The condition that projections form a band (are closed
under multiplication) is shown to be a very natural one. One-sided reducts are
considered, and compared to (generalised) D-semigroups. Finally the special
case when the category is a groupoid is examined.

##  1\. Introduction

The distant aim of this paper is to enrich the study of a groupoid over a skew
lattice of identities, begun by the first author in [6]. But since a skew
lattice possesses two operations, each of which confers a band structure, a
necessary step is to study a groupoid, or more generally a category, over a
band of objects. That is the more immediate aim here, and locates the paper
squarely in the tradition of the Ehresmann-Schein-Nambooripad (ESN) Theorem,
which (in its various forms) asserts equivalence of certain kinds of
categories having extra structure with certain special kinds of semigroups.
This equivalence can help elucidate one kind of structure in terms of the
other; for example, the ESN Theorem has recently been used to design a
relatively efficient enumeration of finite inverse semigroups [17]. We must
take a little further space to summarise the location of the present paper in
this extensive field.

Ehresmann’s use of pseudogroups in describing symmetries in differential
geometry, and Wagner’s and Schein’s similar use of groupoids and inverse
semigroups, are described by Lawson in Sections 1.1–3 of the book [14], with a
history of the ideas behind their respective contributions outlined in Section
4.4 of the same. The interested reader is referred to those sections, to the
extended account of Hollings [11], or to the original works cited therein.
Suffice it to say here that the constructions of the trace product in a
suitable semigroup, and the pseudoproduct in a suitable groupoid, are central
to the success of the Theorem.

Nambooripad’s concern was with regular semigroups, and the constructions in
his works are more general and more complex; see the survey article of
Szendrei [24], especially Section 4. His point of departure was the
description by Miller and Clifford of a completely $0$-simple (regular)
semigroup as a Rees groupoid [18]; he described and formulated the notion of a
bi-ordered set (characterising the set of idempotents, and initiating a whole
field of research which continues today). This he used in conjunction with a
particular kind of ordered groupoid generalising Rees groupoids (the inductive
groupoids in his terminology—a different use of the term than elsewhere), to
formulate his extension of the Theorem, incidentally making it fully
categorical in flavour. Lawson [15] later extended the use of ordered
groupoids to ordered (small) categories, and this kind of generalisation is
also followed in the paper to hand.

All these authors were concerned to explore the conditions which made the
correspondence work; of course there are many “answers” possible. This paper
has the declared aim of tackling this question in the context of the
idempotents or projections forming a band (it will be seen that this condition
arises naturally). Thus the context here is broader than the semilattice of
idempotents of Ehresmann and Schein, but not as broad as Nambooripad’s bi-
ordered sets. The semigroups we consider are a particular class of the
$P$-semiabundant semigroups of Section 1 in Lawson [15], but distinct from the
Ehresmann semigroups of Section 3, which are the main focus of that paper.

It was realised, particularly in work of the ‘York school’, that the field
overlaps (and has been extended to) other semigroup properties, some of which
arose in the theory of semigroup acts; notably, it was refreshed and organised
by Lawson. Its ramifications and notation have been rationalised in the
unpublished but very influential Notes of Gould [8]; Hollings [10] surveyed
its reach and history. Again the reader seeking more detail is referred to
these surveys. Most recently, contributions relevant to the present paper have
been made by Wang [25], Kudryavtseva [13] and El-Qallali [4]. Wang’s paper
[25]—which includes a useful summary of the several versions of ESN
equivalences mentioned above—is the key part of a project designed to achieve
an ESN equivalence for (a generalisation of) orthodox semigroups. The present
paper begins, in contrast, at the category side. We thus circumvent or
postpone some of the arcane semigroup properties otherwise required, such as
generalised Green’s relations and congruence properties. When specialising to
groupoids, however, we reach a similar endpoint to Wang’s, and the
relationship between our approaches is yet to be explored.

Jones’s paper [12] on restriction semigroups is close in spirit to ours on the
semigroup side, though we also consider equivalent categories. Jones
relativises the Ehresmann-style axioms to a specified set $P$ of commuting
projections; in a generalisation, he includes in his scheme the regular
$\ast$-semigroups of Nordahl and Scheiblich [19], in which $P$ is generally
not closed under multiplication. More recently, Branco et al. [2] and Lawson
[16] have focused on the class of Ehresmann semigroups. In contrast to all
these, our $P$ is always a band.

An important distinction among all these studies is between one-sided and two-
sided conditions. Gould and Stokes [9, 21, 22] have addressed corresponding
questions for the one-sided case, in a strand which goes back to Cockett and
Lack’s _restriction categories_ [3]. Because of our concern with categories
and the ESN theorem, our approach is mainly two-sided, though we are able also
to make a small contribution in the one-sided theory in Sections 4 and 6
below. All that said, in the end we aim to understand the structures that
arise when a band or a skew lattice is extended in a manner analogous with
inverse semigroup extensions of a semilattice—or a lattice, as in boolean
inverse semigroups.

##  2\. Small categories

The system $(C,\circ,+,-)$, where $C$ is a set, $\circ$ a partially defined
binary operation $C\times C\rightarrow C$, and $+,-$ are unary maps
$C\rightarrow C$, is a (small) _category_ if it satisfies the axioms

(2.1a) $\displaystyle x^{+}\circ x=x=x\circ x^{-};$ (2.1b)
$\displaystyle(x^{+})^{-}=x^{+},(x^{-})^{+}=x^{-};$ (2.1c) $\displaystyle
x\circ y\text{ is defined precisely when }x^{-}=y^{+};$ (2.1d)
$\displaystyle\text{ when }x^{-}=y^{+},\hskip 6.0pt(x\circ y)^{+}=x^{+}\text{
and }(x\circ y)^{-}=y^{-};$ (2.1e) $\displaystyle(x\circ y)\circ
z=x\circ(y\circ z)\text{ whenever the products are defined}.$

The set $C^{+}=\\{e\in C\colon e=e^{+}\\}=C^{-}$ is the set of _identities_ or
_objects_. Axiom (2.1b) implies
$(x^{+})^{+}=((x^{+})^{-})^{+}=(x^{+})^{-}=x^{+}$ and similarly or dually,
$(x^{-})^{-}=x^{-}$. For $e\in C^{+}$ and $x\in C$, $e\circ x$ is defined if
and only if $x^{+}=e^{-}=e$ and then $e\circ x=x$; and a dual statement holds
for $x\circ e$.

##  3\. Transcription categories

A _transcription category_ is a category as above, endowed with two more maps
$C\times C^{+}\rightarrow C$, called _transcription maps_ and written as
follows: for $e,f\in C^{+}$ and $x\in C$, ${}_{e}|x,x|_{f}$ denote arrows
(members of $C$) satisfying the axioms below, which are based on those for
ordered or inductive categories. Unlike some other studies, we do not impose
any order conditions on the objects to which $x\in C$ may be ‘restricted’:
this is partly why we have used the term ‘transcription’.

(3.1a) ${}_{e}|f=e|_{f};$ (3.1b) ${}_{x^{+}}|x=x=x|_{x^{-}};$ (3.1c)
${}_{e}|(_{f}|x)=\;_{(_{e}|f)}|x\text{ and its dual
}x|_{(e|_{f})}=(x|_{e})|_{f};$ (3.1d) ${}_{e}|(x\circ
y)=(_{e}|x)\circ(_{(_{e}|x)^{-}}|y)\text{ and its dual }(x\circ
y)|_{f}=(x|_{(y|_{f})^{+}})\circ y|_{f};$ (3.1e)
$\displaystyle(_{e}|x)^{+}=\;_{e}|x^{+}\text{ and its dual
}(x|_{f})^{-}=x^{-}|_{f};$ (3.1f)
$\displaystyle(_{e}|x)|_{f}=\;_{e}|(x|_{f}).$

Axiom (3.1a) ensures the two maps agree on $C^{+}\times C^{+}$ and by (3.1e)
map $C^{+}\times C^{+}\rightarrow C^{+}$. Axiom (3.1b) implies ${}_{e}|e=e$
and so with associativity from (3.1c) we have that $C^{+}$ is a band under the
operation $|\,\colon\,(e,f)\mapsto e|_{f}$. Where convenient, we may write
$e|_{f}$ as $e|f$ because of (3.1a). The band $C^{+}$ acts on the set $C$ on
both left and right by (3.1c), interacting with the category operations as per
(3.1d, e), and the two actions are linked by (3.1f).

Observe that the natural-looking equation
$(_{e}|x)^{-}=\;_{(_{e}|x)^{-}}|x^{-}=(_{e}|x)^{-}|_{x^{-}}$ and its dual
$(x|_{f})^{+}=x^{+}|_{(x|_{f})^{+}}=\;_{x^{+}}|(x|_{f})^{+}$ are consequences
of certain of these axioms: put $y=x^{-}$ in (3.1d), so the compositions are
defined and ${}_{e}|x={{}_{e}}|(x\circ
x^{-}){=}(_{e}|x)\circ(_{(_{e}|x)^{-}}|x^{-})$, whence

(3.2)
$(_{e}|x)^{-}\overset{\ref{eq:cat}d}{=}(_{(_{e}|x)^{-}}|x^{-})^{-}\overset{\ref{eq:skewcat}a}{=}{}(_{e}|x)^{-}|_{x^{-}}.$

Given a transcription category $(C,\circ,+,-)$ or $C$ for short, we define a
_pseudoproduct_ $\otimes$ as follows. For $x,y\in C$, define

(3.3) $x\otimes y=(x|_{y^{+}})\circ(_{x^{-}}|y),$

the right hand side being defined in ${C}$ since

$(x|_{y^{+}})^{-}=x^{-}|_{y^{+}}=\>_{x^{-}}|y^{+}=(\;_{x^{-}}|y)^{+}$

by (3.1e) and (3.1a). The following properties will be required. Note that
$\otimes$ extends both $\circ$ and the restriction operations $|$ when they
are defined.

######  Lemma 3.1.

  1. i)

If $x\circ y$ is defined then $x\otimes y=x\circ y$;

  2. ii)

if $e\in C^{+}$ and $x\in C$, $e\otimes x=\,_{e}|x$ and $x\otimes e=x|_{e}$;

  3. iii)

$(x\otimes y)^{+}=(x|_{y^{+}})^{+}$ and $(x\otimes y)^{-}=(_{x^{-}}|y)^{-}$;

  4. iv)

if $e\in C^{+}$ and $x,y\in C$, then $\,{}_{e}|(x\otimes y)=(\,_{e}|x)\otimes
y$;

  5. v)

$x\otimes y=(x|_{x^{-}y^{+}})\circ(\,_{x^{-}y^{+}}|y)$.

###### Proof.

(i) If $x^{-}=y^{+}$, then $x|_{y^{+}}=x|_{x^{-}}=x$, by (3.1b), and dually
${}_{x^{-}}|y=y$; the result follows from the definition in equation (3.3).

(ii) We have $e\otimes
x\overset{\text{def.}}{=}(e|_{x^{+}})\circ(\,_{e^{-}}|x)\overset{\ref{eq:skewcat}a}{=}(\,_{e}|x^{+})\circ(\,_{e}|x)\overset{\ref{eq:skewcat}e}{=}(\,_{e}|x)^{+}\circ(\,_{e}|x)=(\,_{e}|x)$,
and dually.

(iii) This comes from the definition and (2.1d).

(iv) The left-hand expression is $\,{}_{e}|(x\otimes
y)=\,_{e}|((x|_{y^{+}})\circ(\,_{x^{-}}|y))\overset{\ref{eq:skewcat}d}{=}(\,_{e}|x|_{y^{+}})\circ(_{(\,_{e}|x|_{y^{+}})^{-}}|(\,_{x^{-}}|y))$,
where the second factor simplifies, using (3.1c), as follows:

$\,{}_{(\,_{e}|x)^{-}x^{-}y^{+}x^{-}y^{+}}|y\overset{\text{band}}{=}\,_{(\,_{e}|x)^{-}x^{-}y^{+}}|y=\,_{(\,_{e}|x)^{-}x^{-}}|y\overset{\ref{oddone}}{=}\,_{(\,_{e}|x)^{-}}|y.$

The right-hand expression is
$(\,_{e}|x)|_{y^{+}}\circ(\,_{(\,_{e}|x)^{-}}|y)$, and the two are equal.

(v) It is enough to note that $x|_{y^{+}}=(x|_{x^{-}})|_{y^{+}}$, etc., and
use axiom (3.1c) and the definition (3.3). ∎

######  Theorem 3.2.

If ${C}$ is a transcription category then $(C,\otimes)$ is a semigroup.

###### Proof.

We have to show associativity of $\otimes$. First, for any $x,y,z\in C$ such
that $x\circ y$ is defined,

$\displaystyle(x\circ y)\otimes z$ $\displaystyle\overset{\ref{defotimes}}{=}$
$\displaystyle(x\circ y)|_{z^{+}}\circ(_{(x\circ y)^{-}}|z)$
$\displaystyle\overset{\ref{eq:skewcat}d}{=}$
$\displaystyle(x|_{(y|_{z^{+}})^{+}}\circ y|_{z^{+}})\circ(_{(x\circ
y)^{-}}|z)$ $\displaystyle\overset{\ref{eq:cat}e}{=}$ $\displaystyle
x|_{(y|_{z^{+}})^{+}}\circ(y|_{z^{+}}\circ\>_{(x\circ y)^{-}}|z)$
$\displaystyle\overset{\ref{eq:cat}d}{=}$ $\displaystyle
x|_{(y|_{z^{+}})^{+}}\circ(y|_{z^{+}}\circ\>_{y^{-}}|z)$
$\displaystyle\overset{\ref{defotimes}}{=}$ $\displaystyle
x|_{(y|_{z^{+}})^{+}}\circ(y\otimes z)$ $\displaystyle=$ $\displaystyle
x\otimes(y\otimes z),$

since $x^{-}=y^{+}$ and $(y\otimes z)^{+}=(y|_{z^{+}})^{+}$ by Lemma 3.1(iii).
Next, in the general case we have

$\displaystyle(x\otimes y)\otimes z$
$\displaystyle\overset{\ref{defotimes}}{=}$
$\displaystyle(x|_{y^{+}}\circ\;_{x^{-}}|y)\otimes z$
$\displaystyle\overset{\text{above}}{=}$ $\displaystyle
x|_{y^{+}}\otimes((\,_{x^{-}}|y)\otimes
z)\overset{\text{Lem}\ref{lem}\text{(iv)}}{=}x|_{y^{+}}\otimes\,_{x^{-}}|(y\otimes
z).$

Consider the last expression above: we know $(x|_{y^{+}})^{-}=x^{-}|y^{+}$ by
(3.1e) and

$(\,_{x^{-}}|y\otimes z)^{+}=x^{-}|(y\otimes z)^{+}=x^{-}|(y|_{z^{+}})^{+}$

by Lemma 3.1(iii), so

$\displaystyle x|_{y^{+}}\otimes\,_{x^{-}}|(y\otimes z)$ $\displaystyle{=}$
$\displaystyle((x|_{y^{+}})|_{x^{-}|(y|_{z^{+}})^{+}})\circ(\,_{x^{-}|y^{+}|x^{-}}|(y\otimes
z)).$

Now the left factor is

$\displaystyle(x|_{y^{+}})|_{x^{-}|(y|_{z^{+}})^{+}}$ $\displaystyle=$
$\displaystyle x|_{(x^{-}|y^{+})(x^{-}|y^{+})|(y|_{z^{+}})^{+}}$
$\displaystyle=$ $\displaystyle x|_{(x^{-}|y^{+})(y|_{z^{+}})^{+}}$
$\displaystyle=$ $\displaystyle x|_{x^{-}|(y^{+}|(y|_{z^{+}})^{+})}$
$\displaystyle=$ $\displaystyle x|_{(y|_{z^{+}})^{+}}$ $\displaystyle=$
$\displaystyle x|_{(y\otimes z)^{+}},$

the right factor is

$\,{}_{x^{-}|y^{+}|x^{-}|y^{+}}|(y\otimes z)$ $\displaystyle=$
$\,{}_{x^{-}|y^{+}}|(y\otimes z)$ $\displaystyle=$
$\,{}_{x^{-}}|(\,_{y^{+}}|y\otimes z)$ $\displaystyle=$
$\,{}_{x^{-}}|(y\otimes z),$

and finally we have $(x\otimes y)\otimes z=x\otimes(y\otimes z)$.

∎

The semigroup $(C,\otimes)$ is denoted $\mathscr{S}(C)$ and has extra
structure, which we proceed to explore.

##  4\. localisable semigroups

A _unary semigroup_ is an algebra $(S,\cdot,+)$ of signature $(2,1)$ such that
$(S,\cdot)$ is a semigroup, that is, $x\cdot(y\cdot z)=(x\cdot y)\cdot z$,
together with a unary map $+\colon S\rightarrow S$. A _left localisable
semigroup_ is a unary semigroup such that the unary map $+\colon S\rightarrow
S$ satisfies

(4.1a) $\displaystyle x^{+}\cdot x$ $\displaystyle=x,$ (4.1b)
$\displaystyle(x\cdot y)^{+}$ $\displaystyle=(x\cdot y^{+})^{+},\text{ and}$
(4.1c) $\displaystyle x^{+}\cdot y^{+}$ $\displaystyle=(x^{+}\cdot y)^{+}.$

A _right localisable_ semigroup is an algebra $(S,\cdot,-)$ such that $(S^{\rm
opp},\cdot,-)$ is a left localisable semigroup; that is, $(S,\cdot,-)$
satisfies the identities

(4.1d) $\displaystyle x\cdot x^{-}$ $\displaystyle=x,$ (4.1e)
$\displaystyle(x\cdot y)^{-}$ $\displaystyle=(x^{-}\cdot y)^{-},\text{ and}$
(4.1f) $\displaystyle x^{-}\cdot y^{-}$ $\displaystyle=(x\cdot y^{-})^{-}.$

A _localisable_ semigroup is an algebra $(S,\cdot,+,-)$ of signature $(2,1,1)$
such that $(S,\cdot,+)$ is a left localisable and $(S,\cdot,-)$ a right
localisable semigroup, and the unary operations are linked by the identities

(4.1g) $\displaystyle(x^{+})^{-}=x^{+}\text{ and }(x^{-})^{+}=x^{-}.$

These axioms are based on those for restriction semigroups, as well as the
needs of the Theorems to follow. A more detailed comparison with restriction
semigroups and one-sided variants is given in a later Section. For future use,
let us observe that these axioms occur in dual pairs; to reduce repetition, we
will often leave dual statements implicit.

Let $S^{+}=\\{x^{+}:x\in S\\}$, and note that (4.1g) implies
$S^{-}=\\{x^{-}\colon x\in S\\}=S^{+}$ and also
$(x^{+})^{+}=((x^{+})^{-})^{+}=(x^{+})^{-}=x^{+}$. As in restriction
semigroups, members of $S^{+}$ are called _projections_. The next lemma
records some useful relationships.

######  Lemma 4.1.

Let $(S,\cdot,+)$ be a unary semigroup. Then

  1. i)

(4.1a) and c) imply $(S^{+},\cdot)$ is a band;

  2. ii)

(4.1a), b) and c) imply $(x^{+})^{+}=x^{+}$; and

  3. iii)

if $S^{+}$ is a band, (4.1b) implies (4.1c).

###### Proof.

  1. i)

(4.1c) shows that $x^{+}\cdot y^{+}\in S^{+}$; it also implies $x^{+}\cdot
x^{+}=(x^{+}\cdot x)^{+}$ which $=x^{+}$ by (4.1a).

  2. ii)

Now from (4.1b) we have $y^{+}=(y^{+}\cdot y^{+})^{+}=(y^{+})^{+}$.

  3. iii)

Set $x^{+}$ for $x$ in (4.1b) and use $(x^{+}\cdot y^{+})^{+}=x^{+}\cdot
y^{+}$ by the band condition.

∎

This lemma also indicates that it is a natural and interesting condition that
$S^{+}$ be a band (and not necessarily a semilattice). It is of further
interest to point out the following relations between axioms which involve
both unary operations in (4.1).

######  Lemma 4.2.

Let $(S,\cdot,+,-)$ be a semigroup with two unary operations in which (4.1a),
d) and g) hold. Then

  1. i)

(4.1c) and f) together are equivalent to

(4.2) $(x\cdot y^{+})^{-}=x^{-}\cdot y^{+}=(x^{-}\cdot y)^{+};\text{ and}$

  2. ii)

if $S^{+}$ is a band, (4.1b) and e) together imply (4.2).

###### Proof.

  1. i)

Set $y^{+}$ in place of $y$ in (4.1f) and use (4.1g) to obtain the first
equation of (4.2). For the second (and dual) equation of (4.2), put $x^{-}$
for $x$ in (4.1c). In the converse direction, put $y^{-}$ for $y$ in the first
equation of (4.2) to get $(x\cdot(y^{-})^{+})^{-}=x^{-}\cdot(y^{-})^{+}$,
which is (4.1f) by (4.1g); and put $x^{+}$ for $x$ in the second equation of
(4.2) to get the dual (4.1c).

  2. ii)

By Lemma 4.1(ii) and its dual, (4.1c) and f) hold. By part i), (4.2) follows.

∎

######  Lemma 4.3.

Let $(S,\cdot,+,-)$ be a localisable semigroup. Then for all $x,y\in S$,

  1. i)

$x\cdot y=x\cdot y^{+}\cdot x^{-}\cdot y$ and

  2. ii)

$(x^{+}\cdot y)^{-}\cdot y^{-}=(x^{+}\cdot y)^{-}$.

###### Proof.

  1. i)

Now (4.1a) and d), taken with Lemma 4.1(i), show that

$x\cdot y=x\cdot x^{-}\cdot y^{+}\cdot y=x\cdot(x^{-}\cdot y^{+})^{2}\cdot
y=x\cdot x^{-}\cdot(y^{+}\cdot x^{-})\cdot y^{+}\cdot y=x\cdot y^{+}\cdot
x^{-}\cdot y.$

  2. ii)

Putting $x^{+}\cdot y$ for $x$ in (4.1f) we have

$(x^{+}\cdot y)^{-}\cdot y^{-}=((x^{+}\cdot y)\cdot y^{-})^{-}=(x^{+}\cdot
y)^{-},$

as required. ∎

######  Theorem 4.4.

If ${C}$ is a transcription category then $\mathscr{S}(C)$ is a localisable
semigroup.

###### Proof.

Theorem 3.2 proved associativity of $\otimes$, and axiom (4.1g) is (2.1b). We
must verify the remainder of axioms (4.1), and it is enough to prove (4.1a),
b) and c), since the remainder are their duals. For (4.1a) we see
$x^{+}\otimes x=(x^{+}|_{x^{+}})\circ(_{({x^{+}})^{-}}|x)=x^{+}\circ x=x$,
using (3.1b) and (2.1a). For (4.1b), axiom (2.1d) gives

$\displaystyle(x\otimes y)^{+}$ $\displaystyle=$
$\displaystyle((x|_{y^{+}})\circ(_{x^{-}}|y))^{+}$ $\displaystyle=$
$\displaystyle(x|_{y^{+}})^{+},$ while $\displaystyle(x\otimes y^{+})^{+}$
$\displaystyle=$ $\displaystyle((x|_{y^{+}})\circ(_{x^{-}}|y^{+}))^{+}$
$\displaystyle=$ $\displaystyle((x|_{y^{+}})^{+}$ $\displaystyle\text{ also}.$

For (4.1c),

$\displaystyle(x^{+}\otimes
y)^{+}=(x^{+}|_{y^{+}})^{+}=(x^{+}|_{y^{+}})=x^{+}\otimes{y^{+}},$

using Lemma 3.1. ∎

Given a localisable semigroup $(S,\cdot,+,-)$, define a partial binary mapping
or composition $\circ\colon S\times S\rightarrow S$, called the _trace
product_ , and transcription maps defined as follows.

######  Definition 4.5.

(i) The trace product $x\circ y$ is defined exactly when $x^{-}=y^{+}$, and
then

$x\circ y=x\cdot y.$

(ii) Transcription maps for $x\in S,\,e,f\in S^{+}$ are defined by
$\;{}_{e}|x=e\cdot x$ and $x|_{f}=x\cdot f$.

Complementing Theorem 4.4 we have

######  Theorem 4.6.

If ${S}$ is a localisable semigroup, then $\mathscr{C}({S})=(S,\circ,+,-)$ is
a transcription category.

###### Proof.

We first have to verify axioms (2.1a–e). Associativity in $S$ guarantees
(2.1e). By inspection, (2.1a, b) and c) are already (4.1a), d) and g) and
Definition 4.5(i). Now use $x^{-}=y^{+}$ in (4.1b) to obtain (2.1d) as
$(x\circ y)^{+}=(x\cdot y)^{+}=(x\cdot y^{+})^{+}=(x\cdot x^{-})^{+}=x^{+}$,
and dually.

So $\mathscr{C}(S)$ is a category and next we must deal with axioms (3.1). Of
these, (a, b, c) are immediate from Definition 4.5(ii) and associativity of
$S$. For (3.1d) consider the LHS: ${}_{e}|(x\circ y)=e\cdot(x\cdot y)$ with
$x^{-}=y^{+}$. Now check the composition on the RHS is valid:
$(_{(_{e}|x)^{-}}|y)^{+}=((e\cdot x)^{-}\cdot y)^{+}=(e\cdot x)^{-}\cdot
y^{+}=(e\cdot x)^{-}\cdot x^{-}$, using (4.2). By Lemma 4.3(ii), this last
equals $(e\cdot x)^{-}$. Thus the composite $(_{e}|x)\circ(_{(_{e}|x)^{-}}|y)$
is defined, and equals $(e\cdot x)\cdot(e\cdot x)^{-}\cdot y=e\cdot x\cdot y$,
so (3.1d) is satisfied. Wrapping up, (3.1e) is immediate from (4.1c), and
(3.1f) requires only associativity of ${S}$. ∎

Thus ${C}$ and ${S}$ are equivalent; in fact,

######  Theorem 4.7.

$\mathscr{C}(\mathscr{S}(C))=C$ and $\mathscr{S}(\mathscr{C}(S))=S$.

###### Proof.

The constructions use the same base set $S$ and maps $+,-$. Moreover, $\circ$
is a restriction of $\cdot$, and $\otimes$ and $\cdot$ may be identified: for
$x,y\in S$, $x\cdot y=x\cdot y^{+}\cdot x^{-}\cdot y$ by Lemma 4.3(i). Here,
$(x\cdot y^{+})^{-}=(x^{-}\cdot y)^{+}$ by (4.2), whence $x\cdot y=(x\cdot
y^{+})\circ(x^{-}\cdot y)=(x|_{y^{+}})\circ(_{x^{-}}|y)$ by Definition 4.5,
which equals $x\otimes y$ by (3.3). ∎

### Morphisms

Let $(S,\cdot,+,-)$ and $(S^{\prime},\cdot,+,-)$ (in abuse of notation by
overloading the operation symbols) be localisable semigroups. A map
$\phi\colon S\rightarrow S^{\prime}$ is a _morphism of localisable semigroups_
or a _$\pm$ -morphism_ if it preserves the operations in the usual sense,
i.e., it is a semigroup morphism such that $(x\phi)^{+}=(x^{+})\phi$ and
$(x\phi)^{-}=(x^{-})\phi$. Similarly, let $(C,\circ,+,-)$ and
$(C^{\prime},\circ,+,-)$ be transcription categories. A map $\psi\colon
C\rightarrow C^{\prime}$ is a functor (morphism) of transcription categories
if it is a (small) functor in the usual sense (thus $(x\psi)^{+}=(x^{+})\psi$,
etc.) and preserves the transcription maps, i.e.,
$(x|_{e})\psi=(x\psi)|_{e\psi}$, etc.

######  Theorem 4.8.

A map $\phi\colon S\rightarrow S^{\prime}$ is a morphism of localisable
semigroups if and only if it is a functor of transcription categories
$\mathscr{C}(S)\rightarrow\mathscr{C}(S^{\prime})$.

###### Proof.

Let $\phi\colon S\rightarrow S^{\prime}$ be a morphism, so that
$(x\phi)^{+}=(x^{+})\phi$ in $\mathscr{C}(S^{\prime})$. If $x\circ y$ is
defined in $\mathscr{C}(S^{\prime})$, then
$(x\phi)^{-}=x^{-}\phi=y^{+}\phi=(y\phi)^{+}$, so that $x\phi\circ y\phi$ is
defined and $=x\phi\cdot y\phi=(x\cdot y)\phi=(x\circ y)\phi$. If $e\in
S^{+}$, $(x|_{e})\phi=(x\cdot e)\phi=x\phi\cdot e\phi=(x\phi|_{e\phi})$ and
similarly for the right transcription maps. Conversely, if $\phi\colon
C\rightarrow C^{\prime}$ is a functor, $(x\phi)^{\pm}=(x^{\pm})\phi$ and

$\displaystyle(x\cdot y)\phi$ $\displaystyle=$ $\displaystyle(x\otimes y)\phi$
$\displaystyle=$ $\displaystyle[(x|_{y^{+}})\circ(_{x^{-}}|y)]\phi$
$\displaystyle=$ $\displaystyle(x|_{y^{+}})\phi\circ(_{x^{-}}|y)\phi$
$\displaystyle=$ $\displaystyle(x\phi|_{y^{+}\phi})\circ(_{x^{-}\phi}|y\phi)$
$\displaystyle=$ $\displaystyle x\phi\otimes y\phi$ $\displaystyle=$
$\displaystyle x\phi\cdot y\phi,$

whence $\phi$ is a morphism $S\rightarrow S^{\prime}$. ∎

Corresponding to the definition of morphisms, we make the following

######  Definition 4.9.

A _localisable congruence_ $\theta$ on the localisable semigroup
$(S,\cdot,+,-)$ is a congruence of the semigroup $(S,\cdot)$ for which
$(x,y)\in\theta$ implies $(x^{+},y^{+}),(x^{-},y^{-})\in\theta$. For brevity
we also refer to such a congruence as a _$\pm$ -congruence_.

##  5\. Examples and related classes

For the remainder of the paper, we shall (except where emphasis may be needed)
omit the symbol $\cdot$ and instead denote multiplication in the semigroup(s)
by juxtaposition. In the Introduction we remarked on comparisons with
restriction semigroups and Ehresmann semigroups, and here we expand on those
comments.

###  5.1. Restriction semigroups

By definition (see Kudryavtseva [13] for a recent paper which inspired the
present work), $S$ is a _restriction semigroup_ if it satisfies (in the
present notation) axioms (4.1a, c, d, f, g), the _ample_ identities

(5.1) $xy^{+}=(xy)^{+}x,\hskip 8.53581ptx^{-}y=y(xy)^{-},$

and has commuting projections (CP). In effect, in comparison with restriction
semigroups, localisable semigroups dispense with both (CP) and ample
identities.

Incidentally, (CP) may be replaced by the condition that $S^{+}$ be a band:

######  Lemma 5.1.

If both ample identities (5.1) hold in a semigroup $S$ satisfying (4.1a,d,g)
and $S^{+}S^{+}\subseteq S^{+}$, then (CP) holds.

###### Proof.

Let $p,q\in S^{+}$. Then with $x=p=p^{-}$ and $y=q=q^{+}$ in (5.1) we have
$pq=(pq)^{+}p=(pq)p$; with $x=q$ and $y=p$ we have $qp=p(qp)$. Thus
$pq=pqp=qp$. ∎

In this context it is well-known that (4.1b) is a consequence of (5.1):

$\displaystyle(xy^{+})^{+}$ $\displaystyle=^{5.1}$
$\displaystyle[(xy)^{+}x]^{+}$ $\displaystyle=^{4.1c}$
$\displaystyle(xy)^{+}x^{+}$ $\displaystyle=^{CP}$ $\displaystyle
x^{+}(xy)^{+}$ $\displaystyle=^{4.1c}$ $\displaystyle(x^{+}xy)^{+}$
$\displaystyle=^{4.1a}(xy)^{+}.$

The dual statement also holds, so any restriction semigroup, and in particular
any inverse semigroup, satisfies all equations (4.1) and so is a localisable
semigroup. This raises the question of examining localisable semigroups with
(CP).

###  5.2. Ehresmann semigroups

_Left Ehresmann_ semigroups are unary semigroups satisfying (4.1a), (4.1b) and

(5.2) $(x^{+}y^{+})^{+}=x^{+}y^{+}=y^{+}x^{+}$

(Branco et al.[2]; Lawson [16]). Right Ehresmann semigroups are defined
dually, and we say $S$ is _Ehresmann_ if it is left and right Ehresmann and
(4.1g) holds. Note that (5.2) implies (CP). It turns out that localisable
semigroups are precisely a non-commutative version of Ehresmann semigroups.

######  Lemma 5.2.

$(S,\cdot,+)=S$ is left localisable and satisfies (CP) if and only if it is
left Ehresmann.

###### Proof.

Axioms (4.1a) and b) are common to both definitions; so it is sufficient to
show that, in their presence, the conjunction of (CP) and (4.1c) is equivalent
to (5.2). So consider the identity $(x^{+}y)^{+}=(x^{+}y^{+})^{+}$ holding by
(4.1b).

If (4.1c) holds, the left side is $x^{+}y^{+}$ and with (CP) we have (5.2). If
(5.2) holds, the right side of the same identity is $x^{+}y^{+}$ and this
gives (4.1c); en passant we have (CP) also. ∎

######  Corollary 5.3.

$S$ is localisable and satisfies (CP) if and only if it is Ehresmann.

We consider one-sided localisable semigroups further in the next Section. For
extra developments in the topic of one-sided restriction-like semigroups, the
reader may consult Stokes [23].

###  5.3. Monoids

Since any monoid $S$ with identity $1$ may be regarded as a restriction
semigroup, it also is a localisable semigroup. One simply defines
$s^{+}=s^{-}=1$ for all $s\in S$. There is just one projection, and $S$ is
then called a _reduced_ restriction monoid. (There are localisable monoids
with multiple projections: for example, let $S$ be any localisable semigroup
and adjoin a new identity $1$ with $1^{+}=1^{-}=1$.)

Reduced restriction monoids occur as subsemigroups in any localisable $S$: if
$e\in P$, then consider $M_{e}:=\\{x\in S\colon x^{+}=x^{-}=e\\}$. If $x,y\in
M_{e}$, $(xy)^{+}=(xy^{+})^{+}=x^{+}=e$ by (4.1b) and dually, $(xy)^{-}=e$. So
$xy\in M_{e}$ and $M_{e}$ is a subsemigroup of $S$, a localisable semigroup in
its own right, and a reduced restriction monoid. It coincides with the monoid
of endomorphisms of object $e$ in $\mathscr{C}(S)$.

###  5.4. Orthocryptogroups

Any band $S$ is a localisable semigroup when we take $s^{+}=s^{-}=s$. More
generally, let $S$ be an _orthocryptogroup_ , which is to say it is a union of
groups, is orthodox (its idempotents form a band) and its Green’s relation
$\mathscr{H}$ is a congruence. By Theorem II.8.5 (iii) of the monograph [20],
$s^{0}t^{0}=(st)^{0}$ holds for all $s,t\in S$, where, as usual, $s^{0}$
denotes the identity of the subgroup of $S$ which contains $s$. Then, on
setting $s^{+}=s^{-}=s^{0}$, one immediately verifies that (4.1a–g) all hold,
and $S$ is a localisable semigroup.

Notwithstanding all the above, we still need non-trivial and non-artificial
concrete examples of localisable semigroups.

##  6\. One-sided reducts and generalised Green’s relations

In this section, we compare one-sided versions of localisable semigroups,
considered as a class of unary semigroups, with other such classes. Consider a
left localisable semigroup $(S,\cdot,+)$ as defined in (4.1a, b, c), and let a
relation on $S$ be defined thus:

$s\,\widetilde{\leq}_{R}\,t\text{ if }pt=t\text{ implies }ps=s\text{ for all
}p\in S^{+}.$

It is readily seen that $\widetilde{\leq}_{R}$ is reflexive and transitive,
and contains the right division relation ($s=tx$ for some $x$). We record some
results for further reference; note the crucial use of (4.1c),
$(ps)^{+}=ps^{+}$.

######  Lemma 6.1.

The following are equivalent:

  1. i)

$s\,\widetilde{\leq}_{R}\,t$;

  2. ii)

$s^{+}\,\widetilde{\leq}_{R}\,t^{+}$;

  3. iii)

$s^{+}\leq_{R}t^{+}$;

  4. iv)

$s=t^{+}s$.

###### Proof.

Let $p\in S^{+}$. If (i) holds, $pt^{+}=t^{+}$ implies $pt=t$ and in turn
$ps=s$, and $ps^{+}=(ps)^{+}=s^{+}$, so (ii) holds. Also, since
$t^{+}t^{+}=t^{+}$, (ii) implies $t^{+}s^{+}=s^{+}$, which is (iii). Next,
$t^{+}s^{+}=s^{+}$ implies $t^{+}s=s$, so (iii) implies (iv). Finally, if
$t^{+}s=s$ and $pt=t$, then $ps=pt^{+}s=(pt)^{+}s=t^{+}s=s,$ and (i) holds. ∎

Symmetrising $\widetilde{\leq}_{R}$ gives the ‘swung’ Green’s relation
$\widetilde{\mathscr{R}}=\\{(s,t)\colon ps=s\Leftrightarrow pt=t\\}$.

######  Corollary 6.2.

For all $s,t,u\in S$, $s\,\widetilde{\leq}_{R}\,t$ implies
$us\,\widetilde{\leq}_{R}\,ut$, and so $\widetilde{\mathscr{R}}$ is a left
congruence.

###### Proof.

Suppose $s\,\widetilde{\leq}_{R}\,t$; by part (iv) of the Lemma, $t^{+}s=s$.
Consider $(ut)^{+}us=(ut^{+})^{+}us=(ut^{+})^{+}ut^{+}s=(ut^{+})s=us$; this
shows $us\,\widetilde{\leq}_{R}\,ut$, and the rest follows. ∎

Let $E$ be a distinguished set of idempotents of a unary semigroup
$(S,\cdot,+)$, and recall that $S$ is called _weakly left $E$-abundant_
(Gould) or _left $E$-semiabundant_ (Stokes) if every
$\widetilde{\mathscr{R}}$-class contains an element of $E$. It follows that
left localisable semigroups are weakly left $E$-abundant with
$E=\\{s^{+}\colon s\in S\\}=S^{+}$. However, the converse is false:

Example. Let $S$ be the semigroup with zero $0$ generated by idempotents $e,f$
such that $fe=0$, writing $ef=a$ for convenience. The diligent reader will
verify with a little light work that $S=\\{e,f,a,0\\}$, that
$E=E(S)=\\{e,f,0\\}$, and that the assignment $e^{+}=e=a^{+},f^{+}=f,0^{+}=0$
satisfies $y^{+}x=x$ if and only if $y^{+}x^{+}=x^{+}$. Thus
$x\,\widetilde{\mathscr{R}}\,x^{+}$ for all $x\in S$, and so $S$ is weakly
left $E$-abundant. However the corresponding unary operation $x\mapsto x^{+}$
does not make $(S,\cdot,+)$ left localisable, for $E$ is not a band
($ef=a\notin E$).

On generalised D-semigroups. A more stringent condition requires a _unique_
element of $E$ in each $\widetilde{\mathscr{R}}$-class; this gives the
_generalised D-semigroups_ studied by Stokes [22]. This class neither contains
nor is contained by that of left localisable semigroups. Indeed, a generalised
D-semigroup $S$ in which $S^{+}$ is not a band cannot be left localisable. On
the other hand, a band $S$ having an
$\widetilde{\mathscr{R}}={\mathscr{R}}$-class with distinct elements fails the
uniqueness requirement, so $S$ is left localisable (with $s^{+}=s$) but not a
generalised D-semigroup.

The intersection of these two classes merits consideration. Let $S$ be a left
localisable semigroup in which $S^{+}$ is a left regular band. Conditions 1, 2
and 3 of [22] follow from (4.1a,b,c) and condition 4 holds by left regularity
of $S^{+}$. (The notations $x^{+}$ and $D(x)$ are identified.) Now by
Corollary 2.13 of [22], $S$ is a D-semigroup with band of projections,
necessarily left regular. So $S$ also satisfies condition (D3) of [21], namely
$(xy)^{+}x^{+}=x^{+}(xy)^{+}=(xy)^{+}$; the second of these equations is a
consequence of (4.1c) and the first implies that $S^{+}$ is a left regular
band. So the class of left localisable generalised D-semigroups is identified
as the class of D-semigroups with a band of projections or, equivalently, the
class of left localisable semigroups satisfying $(xy)^{+}x^{+}=x^{+}(xy)^{+}$.

Note, however, that the unary projection map in an arbitrary left localisable
semigroup $(S,\cdot,+)$ may be modified to produce a generalised D-semigroup,
as follows. Choose $X\subseteq S^{+}$ to be a cross-section of the equivalence
$\widetilde{\mathscr{R}}$, and for $s\in S$, define $s^{\oplus}$ to satisfy
$s^{\oplus}\in X$ and $s^{+}\,\widetilde{\mathscr{R}}\,s^{\oplus}$. It follows
that $s^{\oplus}s^{\oplus}=s^{\oplus}$ and $s^{\oplus}s^{+}=s^{+}$, whence
$s^{\oplus}s=s^{\oplus}s^{+}s=s^{+}s=s$. That is, $(S,\cdot,\oplus)$ is a
generalised D-semigroup.

##  7\. Other structural features

### An order

We may define an analogue of the natural partial order on a restriction
semigroup:

######  Definition 7.1.

Write $s\unlhd t$ if $s=s^{+}t=ts^{-}$.

Recall that the natural (Mitsch) partial order $\leq_{M}$ on any semigroup $S$
is defined by $s\leq_{M}t$ if $s=t$ or there exist $x,y\in S$ such that
$s=xt=ty=xty$.

######  Lemma 7.2.

The relation $\unlhd$ is a partial order on $S$ which is contained in the
natural partial order $\leq_{M}$, and coincides with it when restricted to the
band $S^{+}$.

###### Proof.

From Definition 7.1 it is clear that $s\unlhd s$ for all $s\in S$. If $s\unlhd
t\unlhd u$, then $s=s^{+}t=s^{+}t^{+}u=(s^{+}t)^{+}u$ (by (4.1)c) $=s^{+}u$,
and dually $s=us^{-}$, whence $s\unlhd u$. Then if $s\unlhd t\unlhd s$ we have
$s=ts^{-}$ and $t=t^{+}s=t^{+}ts^{-}=ts^{-}=s$. It is immediate from the
definitions that $s\unlhd t$ implies $s\leq_{M}t$. When $s=s^{+}=s^{-}$, we
have $s\unlhd t$ if and only if $s=st=ts$, as claimed. ∎

The order $\unlhd$ is the identity on any reduced restriction monoid, and so
differs in general from the Mitsch order.

### Actions and transformation representations

The transcription structures lead to actions of $S$ on the projections. These
give representations of $S$ by transformations, which naturally have weaker
properties than their counterparts in the restriction semigroup setting. The
development here may be compared to that of El-Qallali et al. [5] and Gomes
and Gould [7], which can go much deeper because of the extra conditions.

######  Definition 7.3.

For $p\in S^{+}$ and $s,t\in S$, let us write $p^{s}$ for $(ps)^{-}\in S^{+}$
and dually, ${}^{t}p$ for $(tp)^{+}$.

The ample identities provide a notion of conjugacy (cf. Araújo et al. [1]) and
although those are not available here, the notations $p^{s}$ and ${}^{t}p$
conveniently and compactly reflect the fact that there is structure enough to
yield actions.

######  Lemma 7.4.

For $p\in S^{+}$ and $s,t\in S$, $(p^{s})^{t}=p^{st}$ and
${}^{s}(^{t}p)=\,^{st}p$.

###### Proof.

$(p^{s})^{t}=(p^{s}t)^{-}=((ps)^{-}t)^{-}=((ps)t)^{-}$ from axiom (4.1e), and
so $(p^{s})^{t}=(p(st))^{-}=p^{st}$. The second equation is dual. ∎

Thus we have defined a (right) action $S^{+}\times S\to S^{+}$, $(p,s)\mapsto
p^{s}$. Let us write $s\delta\colon p\mapsto p^{s}$ for $p\in S^{+}$, and from
this define the map $\delta\colon s\mapsto s\delta$. By Lemma 7.4,
$(st)\delta=s\delta\circ t\delta$, so $\delta$ is a semigroup morphism and a
representation of $S$ (as a plain semigroup) in the transformation monoid
$\mathscr{T}_{S^{+}}$.

There is dually a left action $(s,p)\mapsto\;^{s}p=(sp)^{+}$ and corresponding
(anti-) representation $\gamma$. Notationally it is most convenient to write
$\gamma$ and $\gamma s$ as left mappings, $\gamma\colon s\mapsto\gamma s$,
where $\gamma s(p)=\,^{s}p=(sp)^{+}$, so that $\gamma
st(p)=\,^{st}p=(stp)^{+}=\gamma s\circ\gamma t(p)$, and $\gamma$ manifests as
a representation $S\rightarrow\mathscr{T}_{S^{+}}$. Thus the combination of
$\gamma$ and $\delta$ results in a representation $\gamma\otimes\delta\colon
S\rightarrow\mathscr{T}_{S^{+}}\times\mathscr{T}_{S^{+}}$, $s\mapsto(\gamma
s,s\delta)$.

Let us note that $\gamma s^{\pm}(p)=(s^{\pm}p)^{+}=s^{\pm}p$, and dually
$(p)s^{\pm}\delta=ps^{\pm}$. (Here and later we use $\pm$ with the meaning
that it may be replaced in a statement by either $+$ throughout or by $-$
throughout.) Thus $\gamma s^{\pm}=\lambda_{s^{\pm}}$, the inner left
translation on $S^{+}$, and $s^{\pm}\delta=\rho_{s^{\pm}}$, the inner right
translation on $S^{+}$. As usual, we denote the band of inner left [right]
translations of $S^{+}$ by $\Lambda_{0}$ [${\rm P}_{0}$], and the band of
linked pairs of inner translations (bitranslations) $(\lambda_{p},\rho_{p})$
of $S^{+}$ by $\Omega_{0}$. Recall that
$S^{+}\cong\Omega_{0}\leq\Lambda_{0}\times{{\rm P}}_{0}$ via the map
$p\mapsto(\lambda_{p},\rho_{p})$, so that we may identify $S^{+}$ with
$\Omega_{0}$.

The point of these remarks is that the definition of projections on the
semigroup $S(\gamma\otimes\delta)$ by means of $(\gamma
s,s\delta)^{\pm}=(\gamma s^{\pm},s^{\pm}\delta)$ is well-defined if and only
if $(\gamma s,s\delta)=(\gamma t,t\delta)$ implies $(\gamma
s^{\pm},s^{\pm}\delta)=(\gamma t^{\pm},t^{\pm}\delta)$. By the preceding
discussion, this is so if and only if
$(\lambda_{s^{\pm}},\rho_{s^{\pm}})=(\lambda_{t^{\pm}},\rho_{t^{\pm}})$, i.e.,
$s^{\pm}=t^{\pm}$. This leads us to consider the relation $\mu=\mu^{S}$ such
that $(s,t)\in\mu$ if and only if

(7.1) $s^{+}=t^{+},s^{-}=t^{-},(sp)^{+}=(tp)^{+}\text{ and
}(ps)^{-}=(pt)^{-}\text{ for all }p\in S^{+}.$

######  Remark 7.5.

We may write (7.1) as

$(s,t)\in\mu\text{ if and only if }(ps)^{\pm}=(pt)^{\pm}\text{ and
}(sp)^{\pm}=(tp)^{\pm}\text{ for all }p\in S^{+}$

since $(ps)^{+}=ps^{+}=p\rho_{s^{+}}$ by (4.1b), etc.

Now we define a relation $\theta$ on $S$ to be _projection-separating_ if
$p,q\in S^{+}$ and $(p,q)\in\theta$ imply that $p=q$. Clearly, $\mu$ is
projection-separating. It may not be a congruence, but we do have the
following.

######  Lemma 7.6.

Let $\theta$ be a $\pm$-congruence on $S$. Then $\theta$ is projection-
separating if and only if $\theta\subseteq\mu$.

###### Proof.

If $(s,t)\in\theta$, then $(s^{\pm},t^{\pm})\in\theta$, and since $\theta$ is
projection-separating, $s^{\pm}=t^{\pm}$. Moreover,
$((sp)^{\pm},(tp)^{\pm})\in\theta$ for any $p\in S^{+}$, whence
$(sp)^{\pm}=(tp)^{\pm}$. Similarly $(ps)^{\pm}=(pt)^{\pm}$, and $(s,t)\in\mu$
ensues by (7.1).

Conversely, if $\theta\subseteq\mu$ and $(s^{+},t^{+})\in\theta$, then
$(s^{+},t^{+})\in\mu$, and by appropriate choices of $p$ in equation (7.1) we
have $s^{+}s^{+}=t^{+}s^{+}=t^{+}t^{+},$ whence $s^{+}=t^{+}$ and $\theta$ is
projection-separating. ∎

By analogy with the theory of inverse semigroups, we say that the localisable
semigroup $T$ is _fundamental_ if $\mu^{T}$ is the identity relation.

##  8\. The groupoid case: $\ast$-localisable semigroups

Recall that a category $C$ is a _groupoid_ if each morphism $x$ has an inverse
in $C$, denoted by $x^{-1}$, such that $x\circ x^{-1}=x^{+}$ and $x^{-1}\circ
x=x^{-}$. (This is equivalent to $C$ being cancellative and regular in the von
Neumann sense.)

######  Definition 8.1.

A _transcription groupoid_ is a groupoid which is also a transcription
category.

We shall provide a description of the corresponding transcription semigroup
$\mathscr{S}(C)$, and to this end make the following definitions.

######  Definitions 8.2.

$(S,\cdot,{}^{*})$ is a _regular unary semigroup_ if it satisfies

(8.1a) $\displaystyle xx^{*}x$ $\displaystyle=x\hskip 8.53581pt\text{ and }$
(8.1b) $\displaystyle x^{*}xx^{*}$ $\displaystyle=x^{*}$ for all $x\in S$. A
regular unary semigroup $(S,\cdot,{}^{*})$ is _$\ast$ -localisable_ if it also
satisfies (8.1c) $\displaystyle(x(xx)^{*}x)^{*}$ $\displaystyle=x(xx)^{*}x,$
(8.1d) $\displaystyle x(xyy^{*})^{*}$ $\displaystyle=xy(xy)^{*}\hskip
8.53581pt\text{ and }$ (8.1e) $\displaystyle(x^{*}xy)^{*}y$
$\displaystyle=(xy)^{*}xy.$

The following observation is well known.

######  Lemma 8.3.

Let $(S,\cdot,{}^{*})$ be a regular unary semigroup. Then ∗ fixes every
idempotent of $S$ if and only if (8.1c) holds.

###### Proof.

If ∗ fixes every idempotent, then it certainly fixes $x(xx)^{*}x$ since (8.1b)
implies this is an idempotent for every $x\in S$. Conversely, if (8.1c) holds
and if $e^{2}=e$, then

$e^{*}\overset{\eqref{eq:starloc
a}}{=}(ee^{*}e)^{*}=(e(ee)^{*}e)^{*}\overset{\eqref{eq:starloc
c}}{=}e(ee)^{*}e=ee^{*}e\overset{\eqref{eq:starloc a}}{=}e\,.\qed$

######  Lemma 8.4.

Let $(S,\cdot,{}^{*})$ be a $\ast$-localisable semigroup. Then (1) the
identity

(8.2) $x^{**}=x$

holds, and (2) ∗ fixes every $e\in S$ satisfying $(e^{*})^{2}=e^{*}$.

###### Proof.

(1) First we show

(8.3a) $\displaystyle x^{*}x^{**}$ $\displaystyle=x^{*}x$ (8.3b)
$\displaystyle x^{**}x^{*}$ $\displaystyle=xx^{*}\,.$

For (8.3a), we compute

$x^{*}x^{**}\overset{\eqref{eq:starloc
b}}{=}x^{*}(x^{*}xx^{*})^{*}\overset{\eqref{eq:starloc
d}}{=}x^{*}x(x^{*}x)^{*}=x^{*}xx^{*}x\overset{\eqref{eq:starloc
a}}{=}x^{*}x\,,$

using Lemma 8.3 in the third equality, and (8.3b) is proved dually. Thus

$x^{**}\overset{\eqref{eq:starloc
b}}{=}x^{**}x^{*}x^{**}\overset{\eqref{eq:ssa}}{=}x^{**}x^{*}x\overset{\eqref{eq:ssb}}{=}xx^{*}x\overset{\eqref{eq:starloc
a}}{=}x\,,$

which is (8.2).

(2) If $(e^{*})^{2}=e^{*}$, then Lemma 8.3 gives $e^{**}=e^{*}$ and so
$e^{*}=e$ by (8.2). ∎

Since (8.1a) and (8.2) evidently imply (8.1b), we could equally well have used
(8.2) as an axiom in place of (8.1b). Now we can justify the terminology
‘$\ast$-localisable’.

######  Theorem 8.5.

Let $(S,\cdot,{}^{*})$ be a $*$-localisable semigroup and define unary
operations ${}^{+},{}^{-}:S\to S$ by $x^{+}=xx^{*}$ and $x^{-}=x^{*}x$ for all
$x\in S$. Then $(S,\cdot,{}^{+},{}^{-})$ is a localisable semigroup.

Conversely, let $(S,\cdot,{}^{+},{}^{-})$ be a localisable semigroup, assume
that $S$ is regular and suppose there exists an inverse mapping ${}^{*}:S\to
S;x\mapsto x^{*}$ such that $x^{+}=xx^{*}$ and $x^{-}=x^{*}x$ for all $x\in
S$. Then $(S,\cdot,{}^{*})$ is $*$-localisable.

###### Proof.

($\Rightarrow$) From (8.1a), we immediately have $x^{+}x=x=xx^{-}$,
$x^{+}x^{+}=x^{+}$, $x^{-}x^{-}=x^{-}$. From (8.1b), we also get
$x^{*}x^{+}=x^{*}$ and $x^{-}x^{*}=x^{*}$.

In (8.1d), replace $x$ with $xy^{+}$. The left hand side becomes
$xy^{+}(xy^{+}y^{+})^{*}=(xy^{+})^{+}$. The right side becomes
$xy^{+}y(xy^{+}y)^{*}=(xy)^{+}$. Therefore

(8.4) $(xy^{+})^{+}=(xy)^{+}\,.$

A dual argument using (8.1e) gives $(x^{-}y)^{-}=(xy)^{-}$.

Next we have

(8.5) $(x^{+}y)^{+}y=x^{+}y\,.$

Indeed,

$(x^{+}y)^{+}y=x^{+}y(x^{+}y)^{*}y=x^{+}y((x^{+})^{-}y)^{*}y\overset{\eqref{eq:starloc
e}}{=}x^{+}y(x^{+}y)^{*}x^{+}y\overset{\eqref{eq:starloc a}}{=}x^{+}y\,.$

Now we compute

$\displaystyle(x^{+}y^{+})^{*}(x^{+}y^{+})^{*}$
$\displaystyle\overset{\eqref{eq:*-tmp1}}{=}(x^{+}y^{+})^{*}((x^{+}y^{+})^{+}y^{+})^{*}$
$\displaystyle=(x^{+}y^{+})^{*}(x^{+}y^{+})^{+}((x^{+}y^{+})^{+}y^{+})^{*}$
$\displaystyle\overset{\eqref{eq:starloc
d}}{=}(x^{+}y^{+})^{*}((x^{+}y^{+})^{+}y)^{+}$
$\displaystyle\overset{\eqref{eq:*-tmp1}}{=}(x^{+}y^{+})^{*}(x^{+}y)^{+}$
$\displaystyle\overset{\eqref{eq:loc-
tmp1}}{=}(x^{+}y^{+})^{*}(x^{+}y^{+})^{+}$ $\displaystyle=(x^{+}y^{+})^{*}\,.$

Thus by Lemma 8.4, we obtain

(8.6) $\displaystyle(x^{+}y^{+})^{*}$ $\displaystyle=x^{+}y^{+}\,,$ hence
(8.7) $\displaystyle x^{+}y^{+}x^{+}y^{+}$ $\displaystyle=x^{+}y^{+}\,.$

Finally,

$(x^{+}y)^{+}\overset{\eqref{eq:loc-
tmp1}}{=}(x^{+}y^{+})^{+}=x^{+}y^{+}(x^{+}y^{+})^{*}\overset{\eqref{eq:*-tmp2}}{=}x^{+}y^{+}x^{+}y^{+}\overset{\eqref{eq:*-tmp3}}{=}x^{+}y^{+}\,.$

A dual argument gives $(xy^{-})^{-}=x^{-}y^{-}$.

For the converse, we have

$(xy)^{+}=(xy^{+})^{+}=xy^{+}(xy^{+})^{*}=xx^{-}y^{+}(xy^{+})^{*}=x(xy^{+})^{-}(xy^{+})^{*}=x(xy^{+})^{*}\,,$

which is (8.1d), and (8.1e) is proved dually. By Lemma 8.4, we also have
(8.2).

Finally we show that ∗ fixes every idempotent. We first do this for elements
of $S^{+}$:

$(x^{+})^{*}=(x^{+})^{*}x^{++}=(x^{+})^{*}x^{+}=(x^{+})^{-}=x^{+}.$

Now assume $e^{2}=e$. Then $e^{+}=ee^{*}=eee^{*}=ee^{+}$, and so

$e^{*}=e^{*}e^{+}=e^{*}ee^{+}=e^{-}e^{+}=(ee^{+})^{-}=(e^{+})^{-}=e^{+}\,.$

This gives $e\overset{\eqref{eq:starloc
b'}}{=}e^{**}=(e^{+})^{*}=e^{+}=e^{*}$. By Lemma 8.3, (8.1c) holds, completing
the proof. ∎

Hence when we consider $(S,\cdot,^{\ast})$ as a localisable semigroup, we
understand that its projections are given by $x^{+}=xx^{\ast}$ and
$x^{-}=x^{\ast}x$.

Note that for the converse in Theorem 8.5, one must assume some compatibility
between an inverse mapping and the unary operations defining localisability;
regularity of the semigroup is not sufficient. Consider, for instance, the
$2$-element semilattice $S=\\{0,1\\}$ with $0<1$, which has the structure of a
localisable semigroup when $x^{+}=x^{-}=1$ for $x=0,1$. However this does not
derive from the only available inverse mapping, which is $x^{*}=x$; for
$0^{+}=1$ but $00^{*}=0$.

######  Corollary 8.6.

$(S,\cdot,\ast)$ is a $\ast$-localisable semigroup if and only if
$\mathscr{C}(S)$ is a transcription groupoid.

###### Proof.

Let $S$ be a $\ast$-localisable semigroup. By Theorem 4.6, $C=\mathscr{C}(S)$
is a transcription category. For any $x\in C$, $x\circ x^{\ast}$ is defined
and equals $xx^{\ast}=x^{+}$, and similarly $x^{\ast}\circ x=x^{-}$, showing
that $C$ is a groupoid. Conversely, let $C$ be a transcription groupoid; by
Theorem 4.4, $\mathscr{S}(C)$ is a localisable semigroup. Since it is also a
groupoid, the map $x\mapsto x^{-1}$ satisfies the conditions for the converse
part of Theorem 8.5 and so $S$ is $\ast$-localisable. ∎

The following result is Theorem 3 in [6]; for completeness, we give a short
proof using the notation of the present work.

######  Proposition 8.7.

Let $S$ be a $\ast$-localisable semigroup. Then every idempotent is a
projection; in fact, for $x\in S$, $x^{2}=x$ implies $x=x^{+}=x^{-}$.
Consequently, $S$ is orthodox.

###### Proof.

If $x^{2}=x$ then $xxx^{\ast}=xx^{\ast}$ and so $xx^{+}=x^{+}$. Then
$x^{+}=(x^{+})^{-}=(xx^{+})^{-}=x^{-}x^{+}$, by (4.1c) with $y=x^{+}$. Dually
or similarly, $x^{-}=x^{-}x^{+}$; and then $x^{+}=x^{-}$ and
$x=xx^{-}=xx^{+}=x^{+}$ as already seen. ∎

### Remarks

  1. (1)

Proposition 8.7 contrasts with the general situation: if $C$ is merely a
category, $E(C)=\\{x\in E(S)\colon x^{+}=x^{-}\\}.$ (As usual, we denote the
set of idempotents in $S$ or $C$ by $E(S)$ or $E(C)$ respectively.) For let
$x\in S$ with $x=x^{2}$ and $x^{+}=x^{-}$. Then by Definition 4.5, $x\circ x$
is defined in $C$ and $x\circ x=x$ in $\mathscr{C}(S)=C$. In the converse
direction, let $x\circ x=x$ in $C$. Then $x^{+}=x^{-}$ and $x=x\otimes
x=x^{2}$.

  2. (2)

The regular involution ∗ is not necessarily an anti-automorphism of $S$. For
example, let $S$ be a band and set $x^{\ast}=x=x^{+}=x^{-}$ for each $x\in S$;
it is easily verified that $C=\mathscr{C}(S)$ is a transcription groupoid. If
$S$ is not a semilattice then there are $x,y\in S$ such that $xy\neq yx$,
which is equivalent to $(xy)^{\ast}\neq y^{\ast}x^{\ast}$.

## Acknowledgements

The authors thank a referee and the editor for the improvements resulting from
(respectively) their gently pointing out a nonsensical passage and suggesting
additional references. For sharing some of their work before publication, they
thank Mark Lawson and Tim Stokes, who also made helpful comments on drafts of
this work including detecting errors.

## References

  * [1] Araújo, J., Kinyon, M. K., Konieczny, J., Malheiro, A.: Four notions of conjugacy for abstract semigroups. Proc. Roy. Soc. Edinburgh Sect. A. 147, 1169–1214 (2017)
  * [2] Branco, M. J. J., Gomes, M. S., Gould, V.: Ehresmann monoids. J. Algebra 443 349–382 (2015) http://dx.doi.org/10.1016/j.jalgebra.2015.06.035
  * [3] Cockett, J. R. B., Lack, S.: Restriction categories I. Categories of partial maps. Theoret. Comput. Sci. 270, 223–259 (2002)
  * [4] El-Qallali, A.: Congruences on ample semigroups. Semigroup Forum 99, 607–631 (2019)  https://doi.org/10.1007/s00233-018-9988-4
  * [5] El-Qallali, A., Fountain, J., Gould, V.: Fundamental representations for classes of semigroups containing a band of idempotents. Comm. Algebra 36, 2998–3031 (2008)
  * [6] FitzGerald, D. G.: Groupoids on a skew lattice of objects. Art Discr. Appl. Math. # P2.03 (2019)  https://doi.org/10.26493/2590-9770.1342.109
  * [7] Gomes, G.M.S., Gould, V.: Fundamental semigroups having a band of idempotents. Semigroup Forum 77, 279–299 (2008)  https://doi.org/10.1007/s00233-007-9041-5
  * [8] Gould, V.: Notes on restriction semigroups. http://www-users.york.ac.uk/ varg1/restriction.pdf (2010). Accessed 22 August 2020
  * [9] Gould, V., Stokes, T.: Constellations and their relationship with categories. Algebra Universalis 77, 271–304 (2017)
  * [10] Hollings, C.: From right PP monoids to restriction semigroups: a survey. Eur. J. Pure Appl. Math. 2, 21–57 (2009)
  * [11] Hollings, C.: The Ehresmann-Schein-Nambooripad theorem and its successors. Eur. J. Pure Appl. Math. 5, 414–450 (2012)
  * [12] Jones, P. R.: A common framework for restriction semigroups and regular $\ast$-semigroups. J. Pure Appl. Algebra 216, 618–632 (2012)
  * [13] Kudryavtseva, G.: Two-sided expansions of monoids. Internat. J. Algebra Comput. 29, 1467–1498 (2019)
  * [14] Lawson, M. V. Inverse Semigroups: The Theory of Partial Symmetries. World Scientific, River Edge, NJ (1998)
  * [15] Lawson, M. V.: Semigroups and ordered categories I. The reduced case. J. Algebra 141, 422–462 (1991)
  * [16] Lawson, M. V.: On Ehresmann semigroups. Semigroup Forum (2021) https://doi.org/10.1007/s00233-021-10200-22021
  * [17] Malandro, M.E.: Enumeration of finite inverse semigroups. Semigroup Forum 99, 679–729 (2019) https://doi.org/10.1007/s00233-019-10054-9
  * [18] Miller, D. D., Clifford, A. H.: Regular $\mathscr{D}$-classes in semigroups. Trans. Amer. Math. Soc. 82, 270–280 (1956)
  * [19] Nordahl, T. E., Scheiblich, H. E.: Regular $\ast$-semigroups. Semigroup Forum 16, 369–377 (1978)
  * [20] Petrich, M., Reilly, N. R.: Completely Regular Semigroups, Canadian Mathematical Society Series of Monographs and Advanced Texts, Vol. 23. Wiley, New York, NY (1999)
  * [21] Stokes, T.: D-semigroups and constellations. Semigroup Forum 94, 442–462 (2017)
  * [22] Stokes, T.: Generalised domain and E-inverse semigroups. Semigroup Forum 97, 32–52 (2018)
  * [23] Stokes, T.: How to generalise demonic composition. Semigroup Forum 102, 28–314 (2021) https://doi.org/10.1007/s00233-020-10117-2
  * [24] Szendrei, M.B.: Structure theory of regular semigroups. Semigroup Forum 100, 119–140 (2020) https://doi.org/10.1007/s00233-019-10055-8
  * [25] Wang, Y.: Weakly $B$-orthodox semigroups. Period. Math. Hung. 68, 13–38 (2014) https://doi.org/10.1007/s10998-014-0023-6


# Dynamics of a dumbbell satellite under the zonal harmonic effect of an
oblate body

Elbaz I. Abouelmagd1,2, Juan L.G. Guirao3 and Juan A. Vera4 1 Mathematics
Department, Faculty of Science and Arts (Khulais), King Abdulaziz, University,
Jeddah, Saudi Arabia  2 Nonlinear Analysis and Applied Mathematics Research
Group (NAAM) Department of Mathematics, King Abdulaziz University, Jeddah,
Saudi Arabia<EMAIL_ADDRESS><EMAIL_ADDRESS>3 Departamento de
Matemática Aplicada y Estadística. Universidad Politécnica de Cartagena,
Hospital de Marina, 30203–Cartagena, Región de Murcia, Spain.–Corresponding
Author–<EMAIL_ADDRESS>4 Centro Universitario de la Defensa. Academia
General del Aire. Universidad Politécnica de Cartagena, 30720-Santiago de la
Ribera, Región de Murcia, Spain<EMAIL_ADDRESS>
###### Abstract.

The aim of the present paper is to study the dynamics of a dumbbell satellite
moving in a gravity field generated by an oblate body considering the effect
of the zonal harmonic parameter. We prove that the pass trajectory of the mass
center of the system is periodic and different from the classical one when the
effect of the zonal harmonic parameter is non zero. Moreover, we complete the
classical theory showing that the equations of motion in the satellite
approximation can be reduced to Beletsky’s equation when the zonal harmonic
parameter is zero. The main tool for proving these results is the
Lindstedt–Poincare’s technique.

###### Key words and phrases:

Dumbbell satellite, Lindstedt–Poincare’s technique, Zonal harmonic parameter,
Beletsky’s equation.

2010 Mathematics Subject Classification. Primary: 70E17, 70E20, 70E40.
Secondary: 37C27.

##  1\. Introduction

From the end of the sixth decade of the last century, a part of the
mathematical community, has directed its attention to the study the so called
_dumbbell body or satellite_ in central gravity, see for instance Morán [21],
Schechter [25], Brereton and Modi [11], Beletsky [10, 9], Maciejewski et al.
[20]; Kirchgraber et al. [18], Krupa et al. [19], Elipe et al. [15], Burov and
Dugain [12] or Nakanishi et al. [23].

Recall that a _dumbbell body_ is a quite simple structure composed by two
masses connected by a massless rod. It is assume that this object is moving
around a planet whose gravity field is approximated by the field of the
attracting center. In general, the distance between the two points masses is
considered to be much smaller that the distance between the satellite’s center
of mass and the attracting center of mass. Thus, it is common to neglect the
influence of the attitude dynamics on the motion of the center of mass and
treat it as an unperturbed Keplerian one.

Rodnikov [24] studied equilibrium positions of a weight on a cable fixed to a
dumbbell–shaped space station moving along a circular geocentric orbit. This
model is composed by two masses coupled by a weightless rod, while the cable
is weightless and non-stretched. The equations of motion are stated when the
motion is produced in a single plane and the center of mass of the system
moves along a circular geocentric orbit. Moreover, the equilibrium
configurations of the system are obtained and the Lyapunov stability of
configurations for two situations, first when the station is composed of equal
masses, second when masses at the ends of the station are different are
analyzed.

For the “dumbbells–load” system with two unilateral connections, all relative
equilibria on the circular Keplerian orbit were established by Munitsina [22].
Recall that a relative equilibria of the system is a point of the phase space
giving an evolution which is a one–parameter orbit of the action of the
symmetry group of the system. These results were interpreted for studying the
relative equilibria for which both connections are stretched in geometrical
terms. The necessary and sufficient conditions for stability of the relative
equilibria were stated.

Celletti and Sidorenko [14] investigated the dumbbell satellite’s attitude
dynamics, when the center of mass moves on a Keplerian trajectory. They found
a stable relative equilibrium position in the case of circular orbits which
disappears as far as elliptic trajectories are considered. They replaced the
equilibrium position by planar periodic motions and they proved this motion is
unstable with respect to out-of-plane perturbations. They also gave some
numerical evidences of the existence of stable spatial periodic motions.

Burov et al. [13] considered the motion of a dumbbell–shaped body in an
attractive Newtonian central field. They used the Poincare’s theory to
determine the conditions for the existence of families of system periodic
motions depending on the arising small parameter and passing into some stable
radial steady–state motion of the unperturbed problem as the small parameter
tends to zero. They also proved that, each of the radial relative equilibria
generates one family of such periodic motions, for sufficiently small
parameter values. Furthermore, they studied the stability of the obtained
periodic solutions in the linear approximation as well as these solutions were
calculated up to terms of the first order in the small parameter.

Guirao et al. [17] gave sufficient conditions for the existence of periodic
solutions of the perturbed attitude dynamics of a rigid dumbbell satellite in
a circular orbit.

The statement of our main results is the following.

######  Theorem 1.

Consider a dumbbell satellite moving in a gravity field generated by an oblate
body considering the effect of the zonal harmonic parameter $A$. The pass
trajectory of the mass center of the system is periodic and different from the
classical one. If $A$ is equal to zero our solution coincides with the
elliptical classical one.

Finally, considering the motion in the satellite approximation we complete the
classical theory, stating the following result.

######  Theorem 2.

The equations of motion in the satellite approximation can be reduced to
Beletsky’s equation when $A$ is equal to zero.

Note that Theorem 1 generalizes Celletti and Sidorenko [14], Burov and Dugain
[12] and Nakanishi et al. [23] due to oblateness parameter.

The structure of the paper is as follows. In Section 2 we present the model
description, the potential, the kinetic energy and the Lagrangian function of
the system. In Section 3 we present the morphology of the equations of motion
and the equation of the mass center of the system. In Sections 4 and 5 we
respectively provide proof of Theorems 1 and 2. We remark that when $J_{2}=0$
is clear that the dynamics occurs on a plane, however when the coefficient
$J_{2}$ is considered the effects of the gravitational potential are not the
same for planes with different inclinations and a natural question is if there
is an invariant plane for the dynamics. The answer of this fact is positive
and it will be a key point in the proofs of our main result. In the Appendix
we provide a proof of this property.

##  2\. Model description

###  2.1. Hypothesis

We assume that the dumbbell satellite is formed by massless rod of length $l$
with to masses $m_{1}$ and $m_{2}$ placed at its ends. Let consider $c$ the
center of mass of the two masses moving in a gravity field generated by an
oblate body whose mass $m$ having mass center located at $0$ where the
distance between $0$ and $c$ is $r$ and $r\gg l$.

Let us consider the orbital reference frame $c$xy with origin at the
dumbbell’s center, and the polar coordinates of the center are $(r,\theta)$.
While the rotation of the satellite relative to ray $oc$ will be determined by
an angle $\Theta$. Furthermore we denote the reduced mass by $\mu$ and the sum
of the two masses by $m_{s}$ where $\mu=m_{1}m_{2}/m_{s}$ and
$m_{s}=m_{1}+m_{2}$, see Figure 1 for details.

Figure 1. The dumbbell satellite model

Now, we assume that $\underline{r}_{i}$ is the position vector of $m_{i}$ with
respect to $0$. Moreover, let the vector $\underline{n}_{i}$ denotes the
position vector of $m_{i}$ with respect to the center of mass of the dumbbell
satellite, $i\in\\{1,2\\}$ .

Therefore, the magnitudes of the position vectors $\underline{r}_{i}$ are
controlled by

(1) $r_{i}^{2}=r^{2}+n_{i}^{2}+2(-1)^{2-i}n_{i}r\cos\Theta$

where

(2) $n_{i}=m_{3-i}l/m_{s}.$

###  2.2. The potential of the model

From the potential theory, the gravitational potential (any object has axial
symmetry $m_{0}$) experienced by the satellite $m$ will be controlled by (see
Murray and Dermott [16])

(3)
$V=-\dfrac{Gm_{0}m}{r_{0}}\left[1-\sum_{n=2}^{+\infty}J_{n}\left(\dfrac{R}{r_{0}}\right)p_{n}(\sin(\delta))\right]$

where:

  1. (1)

$G$ is the universal constant; $m_{0}$ is the mass of the oblate object and
$m$ is the mass of the satellite;

  2. (2)

$R$ is the mean radius of the oblate object;

  3. (3)

$J_{n}$ is a dimensionless coefficient that characterizes the size of
non–spherical components of the potential;

  4. (4)

$r_{0}$ is the distance between $m_{0}$ and $m$;

  5. (5)

$p_{n}\sin(\delta)$ are the Legendre polynomials of degree $n$;

  6. (6)

$\delta$ denotes the latitude of the satellite.

If the two bodies move in the same plane, then $\delta=0$ and equation (3) can
be written as:

(4)
$V=-\dfrac{Gm_{0}m}{r_{0}}\left[1-\sum_{n=2}^{+\infty}J_{n}\left(\dfrac{R}{r_{0}}\right)p_{n}(0)\right],$

where

$p_{2n}(0)=\dfrac{(-1)^{n}2n!}{2^{2n}(n!)^{2}}$, $p_{2n+1}(0)=0$.

In the present model we shall consider the planar motion, for more details on
it see the Appendix, and take the effect of the zonal harmonic up to $J_{2}$,
hence equation (4) can be rewritten as

$V_{0}=-Gm_{0}m(\displaystyle\frac{1}{r_{0}}+\frac{J_{2}R^{2}}{2r_{0}^{3}}),$

see [1, 2, 3, 4, 5, 6, 7, 8] for more details.

If we assume that $R$ represent the unit of distance, $m_{0}$ is also the unit
mass and denote $J_{2}$ by $A$. We have that the potential experienced by the
masses $m_{1}$ and $m_{2}$ are $V_{1}$ and $V_{2}$ such that

(5) $V_{1}=-Gm_{1}(\displaystyle\frac{1}{r_{1}}+\frac{A}{2r_{1}^{3}}),$

(6) $V_{2}=-Gm_{2}(\displaystyle\frac{1}{r_{2}}+\frac{A}{2r_{2}^{3}}).$

Therefore the total potential $V$ can be written as

(7)
$V=-k(\displaystyle\frac{m_{1}}{r_{1}}+\frac{m_{2}}{r_{2}}+A(\frac{m_{1}}{2r_{1}^{3}}+\frac{m_{2}}{2r_{2}^{3}})),$

where $k=G$ denotes the gravity parameter associated to the oblate body.

###  2.3. The kinetic energy of the model

Let the vectors $\underline{e}_{1}$ and $\underline{e}_{2}$ be an orthogonal
set of unitary vectors with $\underline{e}_{1}$ corresponding to the direction
from $0$ to $c$.

Consider $\underline{i}$ and $\underline{j}$ be another orthogonal set of
unitary vectors such that $\underline{i}$ is a vector in the direction of $x$
axis. Consequently the vectors of the locations $\underline{r}_{i}$ and
associates velocities $\underline{v}_{i}$ of masses $m_{i}$ can be written as

$\underline{r}_{i}=\underline{r}+\underline{n}_{i},$

$\displaystyle\underline{v}_{i}=\frac{d\underline{r}_{i}}{dt},$

where

$\underline{r}=r(\cos\theta\underline{i}+\sin\theta\underline{j}),$

$\underline{n}_{i}=(-1)^{i}n_{i}(\cos\Theta\underline{e}_{1}+\sin\Theta\underline{e}_{2}),$

$\underline{e}_{i}=(-1)^{i}\left(\cos\left(\theta+\dfrac{\pi}{i}\right)\underline{i}+\sin\left(\theta+\dfrac{\pi}{i}\right)\underline{j}\right).$

Therefore, after some calculations, we obtain

(8)
$v_{i}^{2}=\left\\{\begin{array}[]{l}\dot{r}^{2}+r^{2}\dot{\theta}^{2}+n_{i}^{2}(\dot{\theta}+\dot{\Theta})^{2}\\\
-2(-1)^{i}\dot{r}n_{i}(\dot{\theta}+\dot{\Theta})\mathrm{s}\mathrm{i}\mathrm{n}\Theta\\\
+2(-1)^{i}rn_{i}\dot{\theta}(\dot{\theta}+\dot{\Theta})\mathrm{c}\mathrm{o}\mathrm{s}\Theta\end{array}\right\\}.$

Since the kinetic energy of the dumbbell satellite system is

(9) $T=\displaystyle\frac{1}{2}\sum_{i=1}^{2}m_{i}v_{i}^{2}.$

Substituting equation (8) into (9), the kinetic energy can be written in the
form

$T=T_{s}+T_{r}.$

where

(10) $T_{s}=\displaystyle\frac{1}{2}m_{s}(\dot{r}^{2}+r^{2}\dot{\theta}^{2}).$

$T_{r}=\displaystyle\frac{1}{2}\mu l^{2}(\dot{\theta}+\dot{\Theta})^{2}.$

Hence

(11)
$T=\displaystyle\frac{1}{2}m_{s}(\dot{r}^{2}+r^{2}\dot{\theta}^{2})+\frac{1}{2}\mu
l^{2}(\dot{\theta}+\dot{\Theta})^{2}.$

###  2.4. The Lagrangian function of the model

Since the Lagrange’s function is defined by $L=T-V$ from equations (7) and
(11) we get

(12)
$L=\displaystyle\frac{1}{2}m_{s}(\dot{r}^{2}+r^{2}\dot{\theta}^{2})+\frac{1}{2}\mu
l^{2}(\dot{\theta}+\dot{\Theta})^{2}+k(\frac{m_{1}}{r_{1}}+\frac{m_{2}}{r_{2}}+A(\frac{m_{1}}{2r_{1}^{3}}+\frac{m_{2}}{2r_{2}^{3}})).$

Therefore, the equations of motion will be governed by

(13) $\displaystyle\frac{d}{dt}(\frac{\partial
L}{\partial\dot{\chi}})-\frac{\partial L}{\partial\chi}=0,\
\chi\in\\{r,\theta,\Theta\\}.$

##  3\. Equation of motion

###  3.1. Equations of motion for the general case

Substituting equation (12) into (13) when $\chi\in\\{r,\theta,\Theta\\}$ the
equations of motion can be written in the following form

(14) $\begin{array}[c]{l}(m_{s}r^{2}+\mu l^{2})\dot{\theta}+\mu
l^{2}\dot{\Theta}=p_{\theta}=Q,\\\
m_{s}(\displaystyle\ddot{r}-r(\frac{p_{\theta}-\mu
l^{2}\dot{\Theta}}{m_{s}r^{2}+\mu l^{2}})^{2})=-k\text{
}\left\\{\begin{array}[]{l}(\frac{m_{1}(r-n\mathrm{c}\mathrm{o}\mathrm{s}\Theta)}{r_{1}^{3}}+\frac{m_{2}(r+(l-n)\mathrm{c}\mathrm{o}\mathrm{s}\Theta)}{r_{2}^{3}})\\\
+\frac{3}{2}A(\frac{m_{1}(r-n\mathrm{c}\mathrm{o}\mathrm{s}\Theta)}{r_{1}^{5}}+\frac{m_{2}(r+(l-n)\mathrm{c}\mathrm{o}\mathrm{s}\Theta)}{r_{2}^{5}})\end{array}\right\\},\\\
\displaystyle\ddot{\Theta}+\frac{2\dot{r}(\mu
l^{2}\dot{\Theta}-p_{\theta})}{r(m_{s}r^{2}+\mu
l^{2})}=\displaystyle-\frac{k(m_{s}r^{2}+\mu
l^{2})\sin\Theta}{m_{s}lr}((\displaystyle\frac{1}{r_{1}^{3}}-\frac{1}{r_{2}^{3}})+\frac{3}{2}A(\displaystyle\frac{1}{r_{1}^{5}}-\frac{1}{r_{2}^{5}})),\end{array}$

where $Q$ is constant and $n_{1}=n$, $n_{2}=l-n$ while $p_{\theta}$ is a
constant expresses the angular momentum conservation.

###  3.2. Dumbbell’s center of motion

Since $(r,\theta)$ is the coordinate of the dumbbell’s center, therefore the
kinetic energy $T_{s}$ is given by equation (10), while the potential of the
center of mass $V_{s}$ is given by

$V_{s}=-Gm_{s}(\displaystyle\frac{1}{r}+\frac{A}{2r^{3}}).$

Consequently, the Lagrange function $L_{s}$ of the center of mass can be
represented in the form

(15)
$L_{s}=\displaystyle\frac{1}{2}m_{s}(\dot{r}^{2}+r^{2}\dot{\theta}^{2})+Gm_{s}(\frac{1}{r}+\frac{A}{2r^{3}}).$

Substituting equation (15) into (13) with $L=L_{s}$ and
$\chi\in\\{r,\theta\\}$ and taking account that the equations of motion can be
written on the form

$\displaystyle\frac{d}{dt}(\frac{\partial
L_{s}}{\partial\dot{r}})-\frac{\partial L_{s}}{\partial r}=0,$

$\displaystyle\frac{d}{dt}(\frac{\partial
L_{s}}{\partial\dot{\theta}})-\frac{\partial L_{s}}{\partial\theta}=0,$

we state that the motion of dumbbell’s center will be controlled by

(16)
$\begin{array}[c]{l}\displaystyle\ddot{r}-r\dot{\theta}^{2}=-k(\frac{1}{r^{2}}+\frac{3A}{2r^{4}}),\\\
m_{s}r^{2}\dot{\theta}=F$ or $r^{2}\dot{\theta}=h,\\\
\dfrac{1}{2}\dot{r}^{2}-\dfrac{k}{r}-\dfrac{6kA}{r^{3}}=E,\end{array}$

where $F$ is a constant, $h$ is the angular momentum which is constant too,
that can be evaluated by the initial conditions and $E$ is the preservation of
the total energy for the dumbbell’s center.

Let be $r=\displaystyle\frac{1}{u}$ , consequently

(17)
$\displaystyle\frac{d^{2}u}{d\theta^{2}}+u=\frac{k}{h^{2}}(1+\frac{3}{2}Au^{2}).$

(18) $\begin{array}[c]{l}\displaystyle\dot{\theta}=-hu^{2},\\\
\dfrac{1}{2}h^{2}\left(\dfrac{du}{d\theta}\right)^{2}-ku-6kAu^{3}=E.\end{array}$

It is worth mentioning that the system of equations (17) does not represent
only dumbbell’s center motion, it represents too the motion of two–body
problem under the effect of the zonal harmonic motion which can be reduced to
the motion of the classical case when $A=0$. Therefore our results on the
dumbbell’s center of motion can be applied it to the motion of two–body
problem.

Now let us go back to dumbbell’s center motion in which we can be assumed that
this motion follows a Kepler’s type orbit when the effect of oblateness
parameter is absent ($A=0$). Consequently the solution can be written as

$r_{0}=\displaystyle\frac{h^{2}/k}{(1+e_{0}\cos\theta)}$

where $r_{0}=\displaystyle\frac{1}{u_{0}}$ , $e_{0}$ is the orbit eccentricity
such that $0\leq e_{0}<1$, in the framework of elliptic orbits and $\theta$ is
a true anomaly of the center of mass. When $\theta=0,\
u_{0}=\displaystyle\frac{1}{r_{p}}=\frac{k}{h^{2}}(1+e_{0}),\
r_{p}=a(1-e_{0})$ is the pericenter (periapsis) and $a$ is a semi–major axis,
see Figures 2 and 3.

Now we look for solutions in the form $u(\theta,\epsilon)$ under the condition
$0<\epsilon\ll 1$. Since $A=J_{2}$ and $J_{2}\in[1\times 10^{-3},1\times
10^{-6}]$ for the most of celestial bodies then we can replace $\epsilon$ by
A. In addition, this solution must hold the initial conditions

$u(0,\displaystyle A)=\frac{1}{r_{p}},$ $D_{\theta}u(0,A)=0.$

Therefore, we search for straight forward expansion of an asymptotes solution
as a tends to zero in the following form

(19) $u(\theta,A)=u_{0}(\theta)+Au_{1}(\theta)+o(A^{2}).$

The effect of the zonal harmonic of $J_{2}$ is taking account, but the
perturbation due to $J_{2}$ is of order about $10^{-3}$ of the unperturbed
main term $(m_{1}/r_{1})$ or $(m_{2}/r_{2})$. While all other coefficients of
zonal harmonic are about $10^{-6}$ or less. Therefore, it is sufficient from
practical point of view, we take the expansion in equation (19) up to $A$. On
the other hand, $o(A^{2})$ represents the effect of the zonal harmonic $J_{4}$
while our potential does not contain the zonal harmonic $J_{4}$. Consequently
we truncate the expansion in equation (19) up to the linear term $A$. In this
case the leading–order perturbation equations are

$\displaystyle D_{\theta\theta}u_{0}+u_{0}=\frac{k}{h^{2}},$ $\displaystyle
D_{\theta\theta}u_{1}+u_{1}=\frac{3}{2}\frac{k}{h^{2}}u_{0}^{2}.$

Under the conditions $u_{0}(0)=\displaystyle\frac{1}{r_{p}}$ ,
$D_{\theta}u_{0}=0,\ u_{1}(0)=0$ and $D_{\theta}u_{1}=0$, hence the solution
is governed by

$u_{0}=\displaystyle\frac{k}{h^{2}}(1+e_{0}\cos\theta)$

and

$u_{1}=\displaystyle\frac{3k^{3}}{2h^{6}}[1+\frac{1}{2}e_{0}^{2}-(1+\frac{1}{3}e_{0}^{2})\cos\theta+e_{0}\theta\sin\theta-\frac{1}{6}e_{0}^{2}\cos
2\theta].$

Therefore, the general expression of the dumbbell’s center motion up to $o(A)$
will be governed by

(20) $u(\theta,A)=u_{0}(\theta)+Au_{1}(\theta).$

##  4\. Proof of Theorem 1

Since equation (20) represents a solution which contains a secular term that
grows in $\theta$. As a result, the expansion is not uniformly valid in
$\theta$ and breaks down when $\theta=o(A)$, furthermore $Au_{1}$ is no longer
a small correction of $u_{0}$. But convergent series approximation of the
periodic solution can be determined by the continuation method known as the
Lindstedt–Poincare’s technique.

Since equation (17) is a second order differential equation, it describes a
dynamical system in which A is a small parameter. Consequently if $A=0$ the
system will be reduced to a harmonic oscillator which has a solution with
period $T=2\pi/\omega_{0}$ where $\omega_{0}=1$.

The continuation method enables us to construct a periodic solution for $A\neq
0$. If we consider that the angular velocity changes due to the non–linear
terms, the asymptotic solution $u(\theta,\ A)$ and the angular velocity
$\omega$ of the dynamical system can be expanded as

(21)
$\begin{array}[c]{l}u(\theta,A)=u_{0}(\theta)+Au_{1}(\theta)+A^{2}u_{2}(\theta)+\ldots\\\
\omega=1+A\omega_{1}+A^{2}\omega_{2}+\ldots\end{array}$

To construct a uniformly valid solution, we will introduce a stretched
variable $\tau=\omega\theta$, therefore

(22)
$\begin{array}[c]{l}\displaystyle\frac{d}{d\theta}=\omega\frac{d}{d\tau},\\\
\displaystyle\frac{d^{2}}{d\theta^{2}}=\omega^{2}\frac{d^{2}}{d\tau^{2}}.\end{array}$

Substituting equations (22) into (17) we obtain

(23)
$\omega^{2}\displaystyle\frac{d^{2}u}{d\tau^{2}}+u=\frac{k}{h^{2}}(1+\frac{3}{2}Au^{2}).$

Now, under the following conditions

$u(\displaystyle 0,A)=\frac{1}{r_{p}},$ $u_{\tau}(0,A)=0,$
$u(\tau+2\pi,A)=u(\tau,A),$

we insert the series expansion (21) into (23) and equating terms of the same
order in $A$ with keeping the terms up to first order of $A$, we obtain the
following:

  * •

The coefficient of $A^{0}$ gives a homogeneous equation in the form

$\frac{d^{2}u_{0}}{d\tau^{2}}+u_{0}=\frac{k}{h^{2}}$

where$\text{ }u_{0}(0)=\frac{1}{r_{p}}\text{ ,
}\frac{du_{0}(0)}{d\tau}=0\;\text{and}\text{
}u_{0}(\tau+2\pi,A)=u_{0}(\tau,A)$ with a solution

(24) $u_{0g}(\tau)=\frac{k}{h^{2}}(1+e_{0}\cos\tau),$

being

(25) $r=\dfrac{h^{2}/k}{1+e_{0}\cos\tau}.$

  * •

The coefficient of A gives a non–homogeneous equation in the form

(26)
$\displaystyle\frac{d^{2}u_{1}}{d\tau^{2}}+u_{1}=a_{1}+a_{2}\cos\tau+a_{3}\cos
2\tau$

where $u_{1}(0)=0$ , $\displaystyle\frac{du_{1}(0)}{d\tau}=0$ and

$a_{1}=\displaystyle\frac{3k^{3}}{2h^{6}}(1+\frac{1}{2}e_{0}^{2}),$
$a_{2}=\displaystyle\frac{3e_{0}k^{3}}{h^{6}}(1+\frac{2\omega_{1}h^{4}}{3k^{2}}),$
$a_{3}=\displaystyle\frac{3e_{0}^{2}k^{3}}{4h^{6}}$

with a particular solution

$u_{1}(\tau+2\pi,A)=u_{0}(\tau,A)$

$u_{1p}=a_{1}+\displaystyle\frac{1}{2}a_{2}\cos\tau+\frac{1}{2}a_{2}\tau\sin\tau-\frac{1}{3}a_{3}\cos
2\tau.$

This solution contain a secular term $a_{2}\tau\sin\tau/2$, to avoid this term
and the solution becomes periodic we have to equate it coefficient by zero,
hence

$\omega_{1}=-\displaystyle\frac{3k^{2}}{2h^{4}}.$

Therefore the general solution of equation (26) is controlled by

(27)
$u_{1g}=\displaystyle\frac{3k^{3}}{2h^{6}}(1+\frac{1}{2}e_{0}^{2})-\frac{k^{3}}{2h^{6}}(3+e_{0}^{2})\cos\tau-\frac{e_{0}^{2}k^{3}}{4h^{6}}\cos
2\tau.$

Substituting equations (24) and (27) into (21), the general solution of
equation (23) becomes

$u=\displaystyle\frac{k}{h^{2}}(1+k_{1})[1+(\frac{e_{0}-k_{2}}{1+k_{1}})\cos\tau+\frac{k_{3}}{1+k_{1}}\cos
2\tau]$

where

$k_{1}=\dfrac{3Ak^{2}}{4h^{4}}(2+e_{0}^{2}),$
$k_{2}=\dfrac{Ak^{2}}{2h^{4}}(3+e_{0}^{2}),$
$k_{3}=-\frac{Ak^{2}e_{0}^{2}}{4h^{4}},$
$\tau=(1-\frac{3Ak^{2}}{2h^{4}})\theta.$

Therefore

(28) $r=\frac{h^{2}/\overline{k}}{(1+e\cos\tau+\overline{e}\cos 2\tau)},$

with

$\overline{k}=k(1+k_{1}),$ $e=(\frac{e_{0}-k_{2}}{1+k_{1}}),$
$\overline{e}=\dfrac{k_{3}}{1+k_{1}}.$

In short, it is clear that the trajectory of the mass center differs from that
assumed by Celletti and Sidorenko [14], Burov and Dugain [12] and Nakanishi et
al. [23] due to oblateness parameter. Although, this solution is periodic.
While this trajectory is the same as their solutions when the effect of
oblateness is ignored. Since $e_{0}<1$ and $A<<1$ as a result $Ae_{0}^{2}<<1$
is very small. Therefore, if we neglect all terms that include $Ae_{0}^{2}$,
the equation (28) will be reduced to

(29) $r=\displaystyle\frac{h^{2}/\overline{k}}{(1+e\cos\tau)},$

$\displaystyle\overline{k}=k(1+\dfrac{3Ak^{2}}{2h^{4}}),$

$e=e_{0}-\displaystyle\dfrac{3Ak^{2}}{2h^{4}}(1+e_{0}).$

This means that the trajectory of the mass center is elliptic as the classical
case with the decreasing of the elliptical parameter and the eccentricity,
ending the proof.

######  Remark 1.

Taking account the oblateness effect we have proved that the solution is
periodic, see equation (28). While for the small value of the parameter A, we
have elliptical solutions as in the classical case with the decreasing in the
elliptical parameter, see (29).

Thus, there is no discontinuity and the solution varies smoothly as $A$
approaches zero.

Figure 2. Variation in the trajectory of dumbbell’s center when $e_{0}=0.3$
for different values of zonal harmonic parameter. Figure 3. Variation in the
trajectory of dumbbell’s center when $e_{0}=0.8$ for different values of zonal
harmonic parameter.

######  Note 1.

Figures 2 and 3 represent the changes in the trajectory of the dumbbell’s
center corresponding to the changes in the eccentricity and in the zonal
harmonic parameter, here we have considered that $k$ and $h$ are equal to 1.
We denote the curves representing the classical case (the effect of zonal
harmonic is switched off) by (CC). If the effect of zonal harmonic is consider
ignoring all terms with coefficients $Ae_{0}^{2}$ the curves will be denoted
by (EC). Finally, by (PC) we denote the general trajectory.

We observe that all trajectories are quasi–elliptical and the decreasing in
the ellipse parameters is very small for small values of classical
eccentricity $e_{0}$ and the zonal harmonic parameter $J_{2}$, see some cases
of Figure 2. While for some relative large values of the classical
eccentricity $e_{0}$ the decreasing in the ellipse parameter is observed
especially when the parameter of zonal harmonic is assigned by big value.

##  5\. Proof of Theorem 2

We shall start by the deduction of the equations of motion in satellite
approximation. Indeed, substituting equations (1) and (2) into (7), the
approximation of the potential energy can be written as

(30) $\displaystyle V=-k(m_{s}(\frac{1}{r}+\frac{A}{2r^{3}})+\frac{\mu
l^{2}}{2r^{3}}(3\cos^{2}\Theta-1)).$

In this potential we neglect all terms that contain coefficients $(1/r)$ with
power four or more, since $l\ll r$. Therefore the Lagrangian function becomes

(31)
$\begin{array}[c]{l}L=\displaystyle\frac{1}{2}m_{s}(\dot{r}^{2}+r^{2}\dot{\theta}^{2})+\frac{1}{2}\mu
l^{2}(\dot{\theta}+\dot{\Theta})^{2}\\\
+k(m_{s}(\displaystyle\frac{1}{r}+\frac{A}{2r^{3}})+\frac{\mu
l^{2}}{2r^{3}}(3\cos^{2}\Theta-1)).\end{array}$

Substituting equation (31) into (13), the approximation equations of motion
are

(32)
$\begin{array}[c]{l}m_{s}(\displaystyle\ddot{r}-r\dot{\theta}^{2})=-k(m_{s}(\frac{1}{r^{2}}+\frac{3A}{2r^{4}})+\frac{3\mu
l^{2}}{2r^{4}}(3\cos^{2}\Theta-1)),\\\ (m_{s}r^{2}+\mu l^{2})\dot{\theta}+\mu
l^{2}\dot{\Theta}=p_{\theta},\\\ \displaystyle\mu
l^{2}(\ddot{\theta}+\ddot{\Theta})=-\frac{3k\mu
l^{2}}{r^{3}}\cos\Theta\sin\Theta.\end{array}$

Now replacing the independent variable $t$ with the starched variable $\tau$
where $\tau=\omega\theta$ and $r^{2}\dot{\theta}=h$ therefore, it is possible
to write $\dot{\tau}=\Omega(\tau)$ such that

(33)
$\displaystyle\Omega(\tau)=\frac{\omega\overline{k}^{2}}{h^{3}}(1+e\cos\tau+\overline{e}\cos
2\tau)^{2}.$

Hence

(34)
$\begin{array}[c]{l}\displaystyle\frac{d}{dt}=\Omega\frac{d}{d\tau}\vspace{0.5cm},\\\
\displaystyle\frac{d^{2}}{dt^{2}}=\Omega^{2}\frac{d^{2}}{d\tau^{2}}+\Omega\Omega^{\prime}\frac{d}{d\tau},\end{array}$

where $(\cdot)^{\prime}$ means $\displaystyle\frac{d}{d\tau}$.

Inserting equations (34) into (32) and using equation (33), we obtain

(35) $(1+e\cos\tau+\overline{e}\cos
2\tau)\Theta^{\prime\prime}-2(e\displaystyle\sin\tau+2\overline{e}\sin
2\tau)(\frac{1}{\omega}+\Theta^{\prime})+\displaystyle\frac{3k}{\omega^{2}\overline{k}}\cos\Theta\sin\Theta=0.$

Since $\tau=\omega\theta$, we can rewrite equation (35) in the form

(36) $\begin{array}[c]{l}(1+e\cos\omega\theta+\overline{e}\cos
2\omega\theta)\displaystyle\frac{d^{2}\Theta}{d\theta^{2}}-2\omega(e\sin\omega\theta+2\overline{e}\sin
2\omega\displaystyle\theta)(1+\frac{d\Theta}{d\theta})\\\
+\displaystyle\frac{3k}{\overline{k}}\cos\Theta\sin\Theta=0\end{array}$

where

$\omega=1-\dfrac{3Ak^{2}}{2h^{4}},$
$\displaystyle\overline{e}=\frac{Ak^{2}e_{0}^{2}}{4h^{4}},$

and

$\displaystyle\overline{k}=k[1+\dfrac{3Ak^{2}}{4h^{4}}(2+e_{0}^{2})].$

Now, for finishing only remark that equation (36) can be reduced to Beletsky’s
equation, see [9] for more details, if we assume the oblateness effect is not
consider (i.e., $A=0$) obtaining the relation

$(1+e\displaystyle\cos\theta)\frac{d^{2}\Theta}{d\theta^{2}}-2e\sin\theta\frac{d\Theta}{d\theta}+3\cos\Theta\sin\Theta=2e\sin\theta,$

which ends the proof.

## Appendix

Let us introduce now an inertial reference frame $\mathcal{I}(\mathbf{O};$
$\mathbf{E}_{1}$,$\mathbf{E}_{2}$,$\mathbf{E}_{3})$. The coordinates of a
generic vector in this reference system are denoted by
$\mathbf{x=(}x,y,z)_{\mathcal{I}}$ . Recall that we are considering a dumbbell
formed by two material points $\mathbf{M}_{1}$, of mass $m_{1}$ and
$\mathbf{M}_{2}$ of mass $m_{2}$ rigidly connected by a segment of constant
length $l$ and negligible mass mutually attracted by a gravitational potential
due to nearly spherical body $\mathbf{M}$. Recall that the potential is given
by

$\mathcal{V}(\mathbf{x})=-\frac{GM}{\left\|\mathbf{x}\right\|}\left(1-\sum_{n=2}^{\infty}J_{n}\left(\frac{R}{\left\|\mathbf{x}\right\|}\right)^{n}P_{n}\left(\frac{z}{\left\|\mathbf{x}\right\|}\right)\right)$

with $G$ is the gravitational constant, $M$ the mass of the body, and $R$ the
equatorial radius. $P_{n}(u)$ is the Legendre polynomial of degree $n$ and
argument $u$, and the $J_{n}$ are constant coefficients characterizing the
potential of the body $M.$ We can define a rotating frame
$\mathcal{R}(\mathcal{G};\mathbf{e}_{1}$,$\mathbf{e}_{2}$,$\mathbf{e}_{3})$,
with $\mathcal{G}$ be the center of masses of the dumbbell, such that the
unitary vector $\mathbf{e}_{3}$ is directed along the dumbbell towards the
point $\mathbf{M}_{2}$ and $\mathbf{e}_{1}$, $\mathbf{e}_{2}$ are two
orthonormal vectors, perpendicular to $\mathbf{e}_{3}$. In this frame, the
principal moments of inertia $(I_{1},I_{2},I_{3})$ of the dumbbell are

$I_{1}=I_{2}=\mu l^{2},\text{ }I_{3}=0$

with

$\mu=\frac{m_{1}m_{2}}{m_{s}}$

where $m_{s}=m_{1}+m_{2}$ and

$\begin{array}[c]{cc}l_{1}=n=\dfrac{m_{2}l}{m_{s}},\text{
}&l_{2}=l-n=\dfrac{m_{1}l}{m_{s}}\end{array}$

the distances from $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$ to $\mathcal{G}.$

The attitude of the dumbbell is given by two angles, namely nutation $\Theta$
and precession $\phi$. The coordinates of points $M_{1}$ and $M_{2}$ in the
space frame $\mathcal{S}$ are

$\begin{array}[c]{l}\mathbf{M}_{1}\equiv-
l_{1}\left(\sin\Theta\sin\phi,-\sin\Theta\cos\phi,\cos\phi\right)_{\mathcal{S}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \mathbf{M}_{2}\equiv
l_{2}\left(\sin\Theta\sin\phi,-\sin\Theta\cos\phi,\cos\phi\right)_{\mathcal{S}}.\end{array}$

The coordinates of $\mathcal{G}$ respect to the inertial frame $\mathcal{I},$
using cylindrical coordinates are

$\mathcal{G}\equiv(r\cos\theta,r\sin\theta,z)_{\mathcal{I}}.$

Using the Koenig’s Theorem, the Lagrangian of the dumbbell is
$\mathcal{L}\left(\mathbf{P};\mathbf{V}_{\mathbf{P}}\right)$ equal to

$\frac{m_{s}}{2}\left(\left(\frac{dr}{dt}\right)^{2}+r^{2}\left(\frac{d\theta}{dt}\right)^{2}+\left(\frac{dz}{dt}\right)^{2}\right)+\frac{\mu
l^{2}}{2}\left(\left(\frac{d\phi}{dt}\right)^{2}+\left(\frac{d\Theta}{dt}\right)^{2}\sin^{2}\phi\right)-\mathcal{U}(\mathbf{P})$

with
$(\mathbf{P};\mathbf{V}_{P})=\left(r,z,\theta,\Theta,\phi;\frac{dr}{dt},\frac{dz}{dt},\frac{d\theta}{dt},\frac{d\Theta}{dt},\frac{d\phi}{dt}\right),$
and

$\mathcal{U}(\mathbf{P})=\mathcal{V}(\mathbf{x}_{\mathbf{M}_{1}})+\mathcal{V}(\mathbf{x}_{\mathbf{M}_{2}}).$

The coordinates of $\mathbf{x}_{\mathbf{M}_{1}}$ and
$\mathbf{x}_{\mathbf{M}_{2}}$ are

$\begin{array}[c]{c}\mathbf{x}_{\mathbf{M}_{1}}\equiv(r\cos\theta-
l_{1}\sin\Theta\sin\phi,r\sin\theta+l_{1}\sin\Theta\cos\phi,z-l_{1}\cos\phi)_{\mathcal{I}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\mathbf{x}_{\mathbf{M}_{2}}\equiv(r\cos\theta+l_{2}\sin\Theta\sin\phi,r\sin\theta-
l_{2}\sin\Theta\cos\phi,z+l_{2}\cos\phi)_{\mathcal{I}}\end{array}$

and

$\begin{array}[c]{l}\left\|\mathbf{x}_{\mathbf{M}_{1}}\right\|^{2}=r^{2}+z^{2}+l_{1}^{2}-2l_{1}\left(z\cos\phi+r\sin\phi\sin\left(\Theta-\theta\right)\right)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\left\|\mathbf{x}_{\mathbf{M}_{2}}\right\|^{2}=r^{2}+z^{2}+l_{2}^{2}+2l_{2}\left(z\cos\phi+r\sin\phi\sin\left(\Theta-\theta\right)\right)\end{array}.$

The potential of the system has the following expression

$\begin{array}[c]{c}\mathcal{U}(r,z,\Theta-\theta,\phi)=-\left(\dfrac{GMm_{1}}{\left\|\mathbf{x}_{\mathbf{M}_{1}}\right\|}\left(1-\sum_{n=2}^{\infty}J_{n}\left(\frac{R}{\left\|\mathbf{x}_{\mathbf{M}_{1}}\right\|}\right)^{n}P_{n}\left(\frac{z-l_{1}\cos\phi}{\left\|\mathbf{x}_{\mathbf{M}_{1}}\right\|}\right)\right)+\right.\vskip
6.0pt plus 2.0pt minus 2.0pt\text{ }\\\
\left.\dfrac{GMm_{2}}{\left\|\mathbf{x}_{\mathbf{M}_{2}}\right\|}\left(1-\sum_{n=2}^{\infty}J_{n}\left(\frac{R}{\left\|\mathbf{x}_{\mathbf{M}_{2}}\right\|}\right)^{n}P_{n}\left(\frac{z+l_{2}\cos\phi}{\left\|\mathbf{x}_{\mathbf{M}_{2}}\right\|}\right)\right)\right)\end{array}$

### A.1 Hamiltonian expressions

From the expressions of the kinetic energy and the potential, we can derive
the Hamiltonian

$\mathcal{H(}\mathbf{P};T\text{
}\mathbf{V}_{P})=\frac{1}{2m_{s}}\left(P_{r}^{2}+\frac{P_{\theta}^{2}}{r^{2}}+P_{z}^{2}\right)+\frac{1}{2\mu
l^{2}}\left(\frac{P_{\Theta}^{2}}{\sin^{2}\phi}+P_{\phi}^{2}\right)+\mathcal{U}(r,z,\Theta-\theta,\phi)$

with

$\mathcal{(}\mathbf{P};T\text{
}\mathbf{V}_{P})=\left(r,z,\theta,\Theta,\phi;P_{r},P_{\theta},P_{z},P_{\Theta},P_{\phi}\right).$

The angles $\theta$ and $\Theta$ appear only as the difference
$\Theta-\theta$, we can reduce the order of the Hamiltonian by means of the
canonical transformation

$(\,\theta,\Theta-\theta,P_{\theta},P_{\Theta})\rightarrow(\lambda,\psi,P_{\psi}-P_{\lambda},P_{\psi}).$

The new Hamiltonian is

$(A.1)\mathcal{H}=\frac{1}{2m_{s}}\left(P_{r}^{2}+\frac{(P_{\psi}-P_{\lambda})^{2}}{r^{2}}+P_{z}^{2}\right)+\frac{1}{2\mu
l^{2}}\left(\frac{P_{\psi}^{2}}{\sin^{2}\phi}+P_{\phi}^{2}\right)+\mathcal{U}(r,z,\psi,\phi).$

The variable $\lambda$ is cyclic and the momentum $P_{\lambda}$ is a constant
of the motion. The Hamiltonian itself is another integral.

### A.2 Equations of the motion

The Hamiltonian equations of the motion are

$(A.2.1)\begin{array}[c]{cc}\dfrac{dr}{dt}=\dfrac{P_{r}}{\mu},&\dfrac{dP_{r}}{dt}=\dfrac{(P_{\psi}-P_{\lambda})^{2}}{\mu
r^{3}}-\dfrac{\partial\mathcal{U}}{\partial r},\\\
\dfrac{dz}{dt}=\dfrac{P_{z}}{\mu},&\dfrac{dP_{z}}{dt}=-\dfrac{\partial\mathcal{U}}{\partial
z},\\\ \dfrac{d\psi}{dt}=\dfrac{P_{\psi}-P_{\lambda}}{\mu
r^{2}}+\dfrac{P_{\psi}}{\mu
l^{2}\sin^{2}\phi},&\dfrac{dP_{\psi}}{dt}=-\dfrac{\partial\mathcal{U}}{\partial\psi},\\\
\dfrac{d\phi}{dt}=\dfrac{P_{\phi}}{\mu
l^{2}},&\dfrac{dP_{\phi}}{dt}=\dfrac{P_{\psi}^{2}\cos\phi}{\mu
l^{2}\sin^{3}\phi}-\dfrac{\partial\mathcal{U}}{\partial\phi}.\end{array}$

Theorem. _The equations_ (A.2.1) _has an invariant manifold given by_ $z\equiv
0,P_{z}\equiv 0,\phi\equiv\pi/2$ and $P_{\phi}\equiv 0.$

_Proof._ Using the equations

$\begin{array}[c]{ll}\dfrac{dz}{dt}=\dfrac{P_{z}}{\mu},\text{ }\vskip 6.0pt
plus 2.0pt minus
2.0pt&\dfrac{dP_{z}}{dt}=-\dfrac{\partial\mathcal{U}}{\partial z},\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\ \dfrac{d\phi}{dt}=\dfrac{P_{\phi}}{\mu l^{2}},\text{
}\vskip 6.0pt plus 2.0pt minus
2.0pt&\dfrac{dP_{\phi}}{dt}=\dfrac{P_{\psi}^{2}\cos\phi}{\mu
l^{2}\sin^{3}\phi}-\dfrac{\partial\mathcal{U}}{\partial\phi}.\end{array}$

the result is immediate.

The Hamiltonian (A.1) restricted to the invariant manifold is

$\mathcal{H}=\frac{1}{2m_{s}}\left(P_{r}^{2}+\frac{P_{\psi}^{2}}{l^{2}}+\frac{(P_{\psi}-P_{\lambda})^{2}}{r^{2}}\right)+\mathcal{U}_{1}(r,\psi).$

with

$(A.2.2)\begin{array}[c]{c}\mathcal{U}_{1}(r,\psi)=-GM\left[\left(\dfrac{m_{1}}{\sqrt{r^{2}+l_{1}^{2}-2l_{1}r\sin\psi}}+\dfrac{m_{2}}{\sqrt{r^{2}+l_{2}^{2}+2l_{2}r\sin\psi}}\right)+\right.\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\left.R^{2}J_{2}\left(\dfrac{m_{1}}{\left(\sqrt{r^{2}+l_{1}^{2}-2l_{1}r\sin\psi}\right)^{3}}+\dfrac{m_{2}}{\left(\sqrt{r^{2}+l_{2}^{2}+2l_{2}r\sin\psi}\right)^{3}}\right)+O(J_{4})\right]\end{array}$

If $r>>l$ and $R=1$, $M=1$, $k=G$, $J_{2}=A$ we obtain

$(A.2.3)\;\;\mathcal{U}_{1}(r,\psi)=-k\left(m_{s}\left(\frac{1}{r}+\frac{A}{2r^{3}}\right)+\frac{\mu
l^{2}}{2r^{3}}\left(3\cos^{2}\psi-1\right)\right).$

### A.3 The Lagrangian function.

The Lagrangian are

$\mathcal{L}\left(r,\lambda,\psi,\frac{dr}{dt},\frac{d\lambda}{dt},\frac{d\psi}{dt}\right)=\frac{m_{s}}{2}\left(\left(\frac{dr}{dt}\right)^{2}+r^{2}\left(\frac{d\lambda}{dt}\right)^{2}\right)+\frac{\mu
l^{2}}{2}\left(\frac{d\left(\lambda+\psi\right)}{dt}\right)^{2}-\mathcal{U}_{1}(r,\psi)$

and the second order equations of the motion are given by

$(A.3)\begin{array}[c]{l}m_{s}\left(\dfrac{d^{2}r}{dt^{2}}-r\left(\dfrac{d\lambda}{dt}\right)^{2}\right)=-\dfrac{\partial\mathcal{U}_{1}}{\partial
r},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \mu
l^{2}\left(\dfrac{d^{2}\lambda}{dt^{2}}+\dfrac{d^{2}\psi}{dt^{2}}\right)=-\dfrac{\partial\mathcal{U}_{1}}{\partial\psi},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ m_{s}r^{2}\dfrac{d\lambda}{dt}+\mu
l^{2}\left(\dfrac{d\lambda}{dt}+\dfrac{d\psi}{dt}\right)=\text{constant}.\end{array}$

It is clear that if we replace the symbols $\lambda$ by $\theta$ and $\psi$ by
$\Theta$ the equation (A.2.3) is the same of equation (30). Also the system of
equations (A.3) becomes into the system of equations (14) when
$\mathcal{U}_{1}$ is represented by equation (A.2.3) and it is the same of
equation (32) when $\mathcal{U}_{1}$ is represented by equation (A.3).

## Acknowledgements

This work has been partially supported by MICINN/FEDER grant number
MTM2011–22587.

## References

  * [1] E.I. Abouelmagd, S.M. El–Shaboury, _Periodic orbits under combined effects of oblateness and radiation in the restricted problem of three bodies_ , Astrophys Space Sci. 341 (2012), 331–341
  * [2] E.I. Abouelmagd, _Existence and stability of triangular points in the restricted three–body problem with numerical applications_ , Astrophys Space Sci. 342 (2012), 45–53
  * [3] E.I. Abouelmagd, _Stability of the triangular points under combined effects of radiation and oblateness in the restricted three-body problem_ , Earth Moon and Planets 110 (2013), 143–155
  * [4] E.I. Abouelmagd, _Existence and stability of triangular points in the restricted three–body problem with numerical applications_ , Astrophys Space Sci. 342 (2012), 45–53
  * [5] E.I. Abouelmagd, _The effect of photogravitational force and oblateness in the perturbed restricted three-body problem_ , Astrophys Space Sci. 346 (2013), 321–332
  * [6] E.I. Abouelmagd, M.A. Sharaf, _The motion around the libration points in the restricted three-body problem with the effect of radiation and oblateness_ , Astrophys Space Sci. 344 (2013), 321–332
  * [7] E.I. Abouelmagd, H.M. Asiri, M.A. Sharaf, _The effect of oblateness in the perturbed restricted three-body problem_ , Meccanica 48 (2013), 2479–2490
  * [8] E.I. Abouelmagd, M.E. Awad, E.M.A. Elzayat, I.A. Abbas, _Reduction the secular solution to periodic solution in the generalized restricted three-body problem_ , Astrophys Space Sci. 350 (2014), 495–505
  * [9] V.V. Beletsky, _Motion of an artificial satellite about its center of mass_ , Israel program for scientific translations. Jerusalem (1966)
  * [10] V.V. Beletsky and D.V. Pankova, _Connected bodies in the orbit as dynamic billiard_ , Regular Chaot. Dynam. 1 (1996), 87–103
  * [11] R.C. Brereton and V.J. Modi, _On the stability of planar librations of a dumbbell satellite in an elliptic orbit_ , Aeronaut. J. 70 (1966), 1098–1102
  * [12] A. Burov and A. Dugain, _Planar oscillations a vibrating dumbbell–like body in a central field of forces_ , Aeronaut. J. 49(4) (2011), 353–359
  * [13] A. Burov, I.I. Kosenko and H. Troger, _On periodic motions of an orbital dumbbell–shaped body with a cabin elevator_ , Mechanics of Solids 47(3) (2012), 269–284
  * [14] A. Celletti and V. Sidorenko, _Some properties of the dumbbell satellite attitude_ , Celestial Mech. Dyn. Astr. 101 (2008), 105–126
  * [15] A. Elipe, M. Palacios and H. Pretka–Ziomek, _Equilibria of the two–body problem with rigid dumbbell satellite_ , Chaos Soliton $\&$ Fractals 35 (2008), 830–842
  * [16] C.D. Murray and S.F. Dermott, _Solar system dynamics_ , Cambridge University Press, (1999)
  * [17] J.L.G. Guirao, J.A. Vera and B.A. Wade, _On the periodic solutions of a rigid dumbbell satellite in a circular orbit_ , Astrophys Space Sci. 346 (2013), 437–442
  * [18] U. Kirchgraber, U. Manz and D. Stoffer, _Rigorous proof of chaotic behavior in a dumbbell satellite model_ , J. Math. Anal. Appl. 251 (2000), 897–911
  * [19] M. Krupa, A. Steindl and I.I. Troger, _Stability of relative equilibria. Part II: Dumbbell satellites_ , Meccanica 35 (2001), 353–371
  * [20] A.J. Maciejewski , _About attitude motion of a satellite with time dependent moments of inertia_ , Artificial Satellites 17(2-3) (1982), 49–60
  * [21] J.P. Morán, _Effects of plane librations on the orbital motion of a dumbbell satellite_ , ARS Journal 31(8) (1961), 1089–1096
  * [22] M.A. Munitsina, _Relative equilibrium on the circular Keplerian orbit of the “Dumbbells-Load” system with unilateral connections_ , Automation and Remote Control 68(9) (2007), 1476–1481
  * [23] K. Nakanishi, H. Kojima and T. Watanabe, _Trajectories of in plane periodic solutions tethered satellite system projected on van der Pol planes_ , Acta Astronautica 68 (2011), 1024–1030
  * [24] A.V. Rodnikov, _Equilibrium positions of a weight on a cable fixed to a dumbbell–shaped space station moving along a circular geocentric orbit_ , Cosmic Research 44 (2006), 58–68
  * [25] H.B. Schechter, _Dumbbell librations in elliptic orbits_ , AIAA Journal 2 (1964), 1000–1004