An. S t. Univ. Ovidius Constant a [604571]

An. S t. Univ. Ovidius Constant a
Vol. xx (x),201x ,0{24
Monad and its Eilenberg-Moore category on the category of
probabilistic convergence spaces1;;y;2
T. M. G. Ahsanullah1;, Tesnim Meryem Barany, Fawzi Al-Thukair2
Abstract
Motivated by the category of probabilistic convergence spaces – a supercategory of the category of
topological spaces, recently, we brought to light the categories of probabilistic convergence groups,
probabilistic metric probabilistic convergence groups, probabilistic convergence transformation groups
along with their underpinning natural examples. The purpose of this paper is, rst, to establish a
result on the isomorphism between the categories of probabilistic metric groups, and probabilistic
metric probabilistic convergence groups. Secondly, among others, we explore various monads in rela-
tion with probabilistic convergence groups, and probabilistic convergence spaces, and their associated
algebras. In so doing, we consider a product of the categories of probabilistic convergence groups
and probabilistic convergence spaces in an attempt to construct a monad on it such that the cor-
responding category of algebras, so-called Eilenberg-Moore category, is isomorphic to the category
of probabilistic convergence transformation groups. The functor in question here possesses quite a
good number of decent properties, we glimpse at some of those properties. Starting with a particular
adjunction we achieved a monad; and, conversely, given a monad, we obtain an adjunction having
some intimate relationships. Besides Eilenberg-Moore category of algebras that we acquired herein
this text, we investigate the Kleisli category in a bid to look into their natural anity that exhibit
between these two in the light of the present scenario.
1 Introduction
The notion of monad was discovered by R. Godement in 1958 under the name standard construction ,
[16]. The present day standard term monad is due to S. MacLane, [26]. Monad are among the most
Key Words: Probabilistic metric spaces, probabilistic convergence spaces, probabilistic metric groups, probabilistic
convergence groups, probabilistic convergence transformation groups, natural transformations, adjunction, monad, free
algebra, monadic functor, Eilenberg-Moore category, Kleisli category
2010 Mathematics Subject Classication: Primary: 54A20, 54E70; Secondary 54H15, 57S05
Received:
Revised:
Accepted:
0

Monad and its Eilenberg-Moore category 1
omnipresent structure in category theory. Due to its importance since its inception numerous papers
appeared in various types of convergence theories and much beyond, cf. [7, 8, 10, 18, 22, 23, 24, 27, 28].
In [20], G. J ager introduced a topological category, the category of probabilistic convergence spaces,
PCONV , which is Cartesian closed, and has a function space structure and is well bred. In fact,
it is a topological universe. We refer to [5, 6, 11, 30] for the category of convergence spaces, CONV
while for the well-known theory of the category of probabilistic metric spaces, we refer to [4, 9, 13,
14, 15, 29, 32, 33, 34]. In [20], it is shown that the category of probabilistic metric spaces, PMET is
isomorphic to subcategory of PCONV . As a continuation of this theory of probabilistic convergence
spaces, PCONV , we introduced and investigated in [2] and [3], the category of probabilistic convergence
groups and probabilistic convergence transformation groups, respectively. Our interest in this paper is to
look at various monads and their algebras from the perspective of the category of probabilistic convergence
spaces. In an attempt to do that we construct a monad on the product of the categories of probabilistic
convergence groups and probabilistic convergence spaces. We show that the corresponding category of
algebras – so called Eilenberg-Moore category of the monad or the category of T-algebras [1, 12], is
isomorphic to the category of probabilistic convergence transformation groups, deducing some elegant
categorical properties. In order to achieve our goal we look at our paper on probabilistic convergence
transformation groups slight dierently without deviating from the main theme – we introduce some
mappings in Section 5 that are essential ingredients of this paper. In Section 4, we add some results
on the category of probabilistic convergence groups, [2] from categorical perspective showing that the
category of probabilistic metric probabilistic convergence groups, PMPCONVGRP is isomorphic to
the category of probabilistic metric groups, PMETGRP ; later, this result is used to obtain a monad. In
Section 7, we show that starting with an adjunction, we obtain a monad, and conversely, given a monad,
we obtain an adjunction which turned out to be the original monad. In Section 8, brie
y we look at
Kleisli category in relation to our present ndings.
2 Preliminaries
If (A;) is an ordered set, we denote byV
j2Jjthe inmum, whileW
j2Jjdenotes the supremum, if
they exist, of the set fj:j2JgA. In case of a two-point set f;gwe write^and_,
respectively. A function ': [0;1]! [0;1], which is non-decreasing, left-continuous on (0 ;1) and
satises'(0) = 0 and '(1) = 1, is called a distance distribution function [33]. The set of all distance
distribution functions is denoted by
+. For example, for each 0 a<1the functions
a(x) =8
<
:0 if 0xa
1 ifa<x1and1(x) =8
<
:0 if 0x<1
1 ifx=1
belong to
+. The set
+is ordered pointwisely, i.e., for '; 2
+we dene' if for allx0, we
have'(x) (x). The smallest element of
+is then1and the largest element is 0. If'; 2
+,

2 T. M. G. Ahsanullah, T. M. Baran, Fawzi Al-Thukair
then also'^ 2
+. Furthermore, if 'j2
+for allj2J, thenW
j2J'j2
+; for further details cf.
[33].
A binary operation :
+
+!
+, which is commutative, associative, non-decreasing in each place
satisfying the boundary condition (';0) ='for all'2
+, is called a triangle function [33]. The largest
triangle function is the pointwise minimum ('; ) ='^ . It is easy to prove that a triangle function
that is idempotent, i.e., for which (';') ='for all'2
+, must be the largest triangle function.
For a good survey on triangle functions, see e.g. [32]. A triangle function is called continuous [33, 34] if
it is a continuous function with respect to the topology and product topology induced by the modied
L evy metric. A triangle function is called sup-continuous [33, 34] if(W
j2J'j; ) =W
j2J('j; ) for all
'j; 2
+(j2J). For further study on sup-continuity and its relation to continuity, we refer to [33].
For a setS, we denote by P(S) its power set. The set of lters on the set Sis denoted by F(S). We order
this set by set inclusion and we denote for p2Sthe point lter by [ p] =fFS:p2Fg. IfF2F(S)
andG2F(T), then the lters on STgenerated by the sets of the form fFG:F2F;G2Ggis
denoted by FG. If (S;
) is a group and F;G2F(S), then we dene FGas a lter generated by
the setsF
G=fpq:p2F;q2Gg, whereF2FandG2G. The lter F1is generated by the sets
F1=fp1:p2FgforF2F.
For notions of category theory and some of their related results that are use in this paper, without
reference, are taken from Ad amek et. al. [1] and [26]. Let SET be the category of sets and functions
while GRP be the category of groups and group homomorphisms. If Cis any category, then we often
denote the objects of CbyjCjand the morphism by Mor (C). In this paper, we denote any algebraic
operation by
including functions composition, unless otherwise mentioned.
3 Category of probabilistic convergence spaces
Denition 3.1. [20] LetS2jSETj. A family of mappings ( c':F(S)!P(S))'2
+which satises the
axioms
(PC1)p2c'([p]),p2S,'2
+;
(PC2) if FG, thenc'(F)c'(G),8F;G2F(S) and8'2
+;
(PC3)' ,c (F)c'(F),8F2F(S),8'; 2
+;
(PC4)p2c1(F),8p2S,8F2F(S),
is called a probabilistic convergence structure onS. The pair
S;c= (c')'2
+

is called a (distance
distribution function indexed) probabilistic convergence space . If (S;c) satises further the axiom
(PC5)c'(F)\c'(G)c'(F^G);8'2
+,8F;G2F(S),
then we speak of a probabilistic limit space .
A mapping f: (S;c)!
S0;c0

between probabilistic convergence spaces (resp. between probabilistic
limit spaces) is called continuous if8p2S,8F2F(S),p2c'(F))f(p)2c0
'(f(F)).
PCONV (resp. PLIM ) denotes the category of probabilistic convergence (resp. limit) spaces.

Monad and its Eilenberg-Moore category 3
Example 3.2. LetS2jSETjand consider the probabilistic convergence structure on Sas given by
p2c1(F),8F2F(S),p2S
Then the pair ( S;c) is called an indiscrete probabilistic convergence space.
Example 3.3. LetS2jSETjand consider the probabilistic convergence structure on Sas given8F2
F(T),'2
+andp2Tby
p2c'(F)()F[p], andp =2c1(F), otherwise.
Then the pair ( S;c)is called a discrete probabilistic convergence space.
Denition 3.4. [33] A probabilistic metric space is a pair (S;F), whereS2jSETj, andF:SS!
+
satisfying the following:
(FM1)F(p;q) =0,p=q;
(PM2) F(p;q) =F(q;p);
(PM3)(F(p;q);F(q;r))F(p;r)
with respect to a triangle function ; usually, we denote F(p;q) byFpq.
A mapping f: (S;F)!(S0;F0) between probabilistic metric spaces is called non-epansive if for all
p;q2S,FpqF0
f(p)f(q).
The category of probabilistic metric spaces and non-empansive mappings is denoted by PMET . We
recall Tardi's neighborhood system as described in [34]. Let ( S;F)2jPMETj. Given'2
+,>0
andp2S, the (';)-neighborhood ofpis dened by
N';
p=fq2S:Fp;q(x+) +'(x);x2[0;1
g:
Then the lter N'
pis given by N'
p= [fN';
p:>0g].
Now one can dene p2cF
'(F)()FN'
p, cf. [20] for further details.
Lemma 3.5. [20] If (S;F)2jPMETj, then
S;cF= (cF
')'2
+
2jPCONVj.
Theorem 3.6. [20, 1] The category PCONV is topological over SET with respect to the forgetful functor
U:PCONV!SET . Moreover, PCONV is Cartesian closed, and extensional.
Denition 3.7. [20] A probabilistic convergence space
S;c= (c')'2
+

is called
(a)probabilistic pretopological space if for allp2S,'2
+and for all F2F(S),
p2c'(F) if and only if FN'
p, where N'
pis dened as '-neighborhood of pby:
N'
p=V
p2c'(F)F.

4 T. M. G. Ahsanullah, T. M. Baran, Fawzi Al-Thukair
(b)T1-space if for allp;q2S,W
':p2c'([q])'=0impliesp=q.
(c)symmetric if for all'2
+and for allp;q2S:q2c'([p]) impliesp2c'([q]).
(d)-transitive if for allp;q;r2Sand for all '; 2
+:p2c'([q]) andq2c'([r]) implyp2
c('; )([r]).
Denition 3.8. [20] A probabilistic convergence space
S;c= (c')'2
+

is said to satisfy the axiom
(PM) if for all p2S, for all'2
+and for all ultralters U2U(S) the following holds:
p2c'(U),8U2U; > 09q2U s:tW
:p2c ([q]) (x+) +'(x)8×2[0;1
):
Denition 3.9. [2] Let
S;c= (c')'2
+

be a probabilistic convergence space and consider H(S) =
ff:S!S,fis a homeomorphism, i.e., fis bijective and both f, andf1are continuousg. Dene
a probabilistic convergence structure cc
'onH(S), the group of homeomorphisms under composition:
2F(H(S)) andf2H(S) we dene
f2cc
'()() 8p2S; p2cS
(F))f(p)2c (ev(F)); ';
8p2S; p2cS
(F))f1(p)2cS

ev(1F)

; ',F2F(S),
where 1= [f1:2g], and1=ff12H(S):f2H(S)g.
4 Category of probabilistic convergence groups
Denition 4.1. [2] A triple
S;
;c= (c')'2
+

is called a probabilistic convergence group under a
triangular function if
(PCG1) (S;
)2jGRPj;
(PCG2) (S;c)2jPCONVj;
(PCM)pq2c('; )(FG), whenever p2c'(F), andq2c (G) ;
(PCI)p12c'(F1) whenever p2c'(F).
Furthermore, if in (PCG2) ( S;c)2 jPLIMjis a probabilistic limit space, then the resulting triple

S;
;c= (c')'2
+

is called a probabilistic limit group under triangular function .
The category of probabilistic convergence (resp. limit) groups and continuous group homomorphisms
under the triangle function is denoted by PCONVGRP (resp. PLIMGRP ).
Proposition 4.2. If
Sj;
;(cj
')'2
+

j2Iis a family of probabilistic convergence groups under the triangle
function, then the productQ
jSj;
;c= (c')'2
+
is also a probabilistic convergence group under the
triangle function .
Proof. We need to show thatQ
jSj;
;c
is a product in PCONVGRP . In Theorem 3.4 [20], take
S=QSjthe direct product of a family of groups, and pri:Q
jSj!Si, the projection mappings for
eachi2I. Then clearly for each i2I,priis a surjective group homomorphism and hence yields a source

Monad and its Eilenberg-Moore category 5
S=fpri: (S;
)!(Si;
;(ci
')'2
+)g.
In view of Theorem 3.4 [20] and Proposition 10.53 [1], one can easily prove that Sis initial in PCON-
VGRP, andjSjis a product in GRP .
Example 4.3. Let (T;
) be a group. Then with the indiscrete probabilistic convergence structure as
given by
p2c1(F),8F2F(T), andp2S.
gives the triple ( T;
;c) a probabilistic convergence group, called indiscrete probabilistic convergence group .
Example 4.4. Let (T;
) be a group equipped with a discrete probabilistic convergence structure as given
8F2F(T),'2
+andp2Tby
p2c'(F)()F[p], andp =2c1(F), otherwise,
gives rise to the triple ( T;
;c) a probabilistic convergence group, called discrete probabilistic convergence
group . In fact, one can easily check that the pair ( T;c) is a probabilistic convergence space. The other
remaining conditions can be seen from the following observations: if p;q2TandF;G2F(T), then for
p2c'(F) andq2c (G) yield that F[p] and G[q]. Since FG[p][q] = [pq], along with the
fact that('; )'; implyingp2c('; )(F) andq2c('; )(G), we havepq2c('; )(FG). Now if
p2c'(F), then F[p] which implies F1[p]1= [p1]. Hencep12c'(F1). Thus one obtains the
triple (T;
;c) a probabilistic limit group or a discrete probabilistic limit group.
Theorem 4.5. The category PCONVGRP is topological over GRP with respect to the forgetful
functor PCONVGRP W!GRP , whereas PCONVGRP is not topological over PCONV .
Proof. See for instance Proposition 4.3 [2].
Example 4.6. Consider the concrete category C:=PCONVGRP overX:=GRP , then the discrete
objects are the probabilistic convergence groups with discrete probabilistic convergence structure. In
fact, iff:C!Dis a group homomorphism, then for any p2CandF2F(C),p2c'(F) if and only
ifF[p] if and only if f(F)f([p]) = [f(p)] if and only if f(p)2c'(f(F)). This implies that fis aC-
morphism which means a continuous group homomorphism. Obviously, this is not the case if we consider
the concrete category PCONVGRP overPCONV . In fact, the morphisms in PCONV are continuous
mappings while the morphisms in PCONVGRP are continuous group homomorphisms. Remark that
if we consider PCONVGRP as a concrete category over GRP , then the indiscrete objects are the
probabilistic convergence groups with indiscrete probabilistic convergence structure.
Theorem 4.7. [3] Letbe the largest triangle function, and c= (c')'2
+be a probabilistic conver-
gence structure (resp. probabilistic limit structure) on S, then
H(S);
;c= (cc)'2
+

is a probabilistic
convergence group (resp. probabilistic limit group) under the largest triangle function .

6 T. M. G. Ahsanullah, T. M. Baran, Fawzi Al-Thukair
Denition 4.8. [3] A triple ( S;
;F) is called a probabilistic metric group under triangle function 
if (S;
) is a group and ( S;F) is a probabilistic metric space under the triangle function such that
the mapping m: (SS;F
F)!(S;F), (p;q)7!pqand|: (S;F)!(S;F),p7!p1are
non-expansive, where the product structure as given below is due to Tardi [34]:
F
F((p1;p2);(q1;q2)) =(Fp1;q1;Fp2;q2), for all (p1;p2);(q1;q2)2SS.
It follows from Lemma 6.1[3] that probabilistic metric groups under triangle function are precisely groups
endowed with an invariant probabilistic metric, i.e., F(ps;qs )Fp;qandF(sp;sq )Fp;q8p;q;s2S;
in fact, invariance implies: Fps;qs =Fp;qandFsp;sq =Fp;q. We denote the category PMETGRP 
consisting of all probabilistic metric groups under a continuous triangle functions as objects and non-
expansive group homomorphisms as morphisms.
Example 4.9. [2] Every probabilistic metric group under a continuous triangle function gives rise to
a probabilistic convergence group.
Given a continuous triangle function ,PMPCONVGRP denotes a subcategory of the category
of probabilistic convergence group, PCONVGRP with objects probabilistic convergence groups and
continuous group-homomorphisms under sup-continuous and continuous triangle functions that are pre-
topological, symmetric, -transitive, T1 and satisfy the axiom (PM).
In view of the Theorem 8.2 [2] in conjunction with the Lemma 6.7 [20], we deduce the following
Lemma 4.10. There exists a functor
:8
>><
>>:PMPCONVGRP !PMETGRP 
(S;
;c= (c')'2
+)7! (S;
;Fc)
f7! f
Theorem 4.11. [2] If (S;
;F)2jPMETGRP j, then
S;
;cF
2jPMPCONVGRP j. Conversely,
if(S;
;c)2jPMPCONVGRP j, then (S;
;Fc)2jPMETGRP j. Furthermore, there are functors
PMETGRP  PMPCONVGRP 

such that  =idPMPCONVGRP and  =idPMETGRP . That is, the categories PMETGRP and
PMPCONVGRP are isomorphic; more generally, theses categories are equivalent.
Proof. In view of the Theorems 6.6, 8.2 in [2] in conjunction with Lemmas 6.9, 6.10 and Theorem 6.11
in [20], we only proof here the missing parts. Let ( S;
;c)2jPMPCONVGRP j. Then
 (S;
;c) =  (S;
;Fc) =
S;
;cFc
= (S;
;c):

Monad and its Eilenberg-Moore category 7
On the other hand, let ( S;
;F)2jPMETGRP j. Then
 (S;
;F) = 
S;
;cF
= (S;
;c) = (S;
;Fc)
Theorem 4.12. Each of the following forgetful functor has adjoint.
(a)PCONVU!SET
(b)PCONVGRP V!SET
(c)PCONVGRP W!GRP .
Proof. (a) follows from the fact that the forgetful functor U:PCONV!SET is a topological functor
and therefore it has adjoint. As for (b), note the fact that each set Swith the identity morphism sends to
an indiscrete probabilistic convergence group ( S;
;c1). In view of the Proposition 21.12 [1] and Theorem
4.5, item (c) has adjoint.
5 Actions of probabilistic convergence groups on probabilistic convergence
spaces
Let
T;
;t= (t')'2
+

2jPCONVGRP j, and
S;
;c= (c')'2
+

2jPCONVj. Let us consider the
mappings denoted by eT
S:S!TSandmT
S:T(TS)!TS, and dened as:
eT
S(s) := (e;s) and mT
S(t;(u;s)) := (tu;s)
respectively, for all s2S,t;u2T. andeis the identity element of the group ( T;
)
Denition 5.1. Anaction of a probabilistic convergence group
T;
;t= (t')'2
+

under the triangle
functionon a probabilistic convergence space
S;c= (c')'2
+

is a continuous mapping :TS!S
(usually denoted by ( t;s)7!ts) subject to the requirements that the following conditions are fullled:
(PCTG1)
mT
S=
(idT), i.e.,(tu;s) =(t;(u;s)), for allt;u2Tands2S;
(PCTG2)
eT
S=idS, i.e.,(e;s) =s, for alls2S.
Denition 5.2. A triple
T;
;t= (t')'2


;
S;c= (c')'2
+

;

( or in short ( T;S; )) is called a
probabilistic convergence transformation group , where
T;
;t= (t')'2
+
2jPCONVGRP junder the
triangle function , and
S;c=
c')'2
+


2jPCONVjwithan action of probabilistic convergence
group on a probabilistic convergence space
S;c= (c')'2
+

. HereTis called phase group whileSis
called phase space of the probabilistic convergence transformation group ( T;S; ).
If we denote t(s) :=(t;s) =:s(t), fort2T,s2S, then the continuous mappings t:S!Sand
s:T!Sare called transitions and motions , respectively of the action .

8 T. M. G. Ahsanullah, T. M. Baran, Fawzi Al-Thukair
Remark 5.3. If (T;S; ) is a probabilistic convergence transformation group, then (t;s) is denoted by
t
sorts. The statement ( T;S; )is a probabilistic convergence transformation group may be rephrased
asTacts onSbyorSis aT-space with action . We sometimes may use theT-space (T;S; ):
Lemma 5.4. Let
T;
;t

2jPCONVGRP j,(S;c)2jPCONVj, and:TS!Sbe a continuous
mapping. Then is an action if and only if the following are true:
(a)e=idS;
(b)tu=tu,
for allt;u2T.
Proof. Let:TS!Sbe an action. Then using (PCTG2) for any s2S,e(s) =(e;s) =
eT
S(s) =
idT(s);i.e.,e=idTwhich is (a). Next, upon using (PCTG1), we have for any s2S,
tu(s) =(tu;s) =
mT
S(t;(u;s))

=
mT
S(t;(u;s)) =
(idT)(t;(u;s)) =(t;(u;s)) =tu(s);
that is,tu=tu, which is (b).
Conversely, if (a) and (b) hold, then as is already continuous, we just need to check items (PCTG1)
and (PCTG2) are satised. To this end, using (a), for any s2S,

eT
S(s) =(e;s) =e(s) =idS(s)
which is (PCTG2). As for (PCTG1), upon using (b), for any t;u2Tands2S, we have

mT
S(t;(u;s)) =(tu;s) =tu(s) =tu(s) =t((u;s)) =(t;(u;s)) =
(idT)(t;(u;s));
that is,
mT
S=
(idT), which is (PCTG1). Hence is an action.
Lemma 5.5. Let(T;S; )be a probabilistic convergence transformation group
withm:TT!Tas the group multiplication in T; in addition, ststands for(t;s), then for any
(t;s)2TS, the following holds:
(i)t
s=s
mt;
(ii)s
mt=ts
Proof. For anyu2T, we have
t
s(u) =t((u;s)) =(t;(u;s)) =s(tu) =s(mt(u)) =s
mt(u)
that is,t
s=s
mt. Now for (ii), we have for any u2T,
s
mt(u) =s(ut) =(ut;s) =uts=ts(u)
that iss
mt=ts.

Monad and its Eilenberg-Moore category 9
Theorem 5.6. [3] Let(';') =',8'2
+, i.e.,is the largest triangle function. Then the
triple
(T;
;d);(S;c);

is a probabilistic convergence transformation group on a probabilistic convergence
space (S;
;c)with respect to if and only if ~: (T;d)!(H(S);c)is continuous homomorphism, where
~(t) =t:
Corollary 5.7. Let(T;S; )be a probabilistic convergence transformation group under the largest triangle
function. Then each mapping t:S!Sis a homeomorphism for t2T. Furthermore, the mapping
~:T!H(S),~7!t, where H(S)the group of homeomorphisms of S, is a morphism between groups
TandH(S)the group of homeomorphisms.
Proof. This is an immediate consequence of the preceding Theorem 5.6.
Remark 5.8. It follows from the Corollary 5.7 that for every probabilistic convergence transformation
group (T;S; ),can be considered as an action of the group Tdon the probabilistic convergence space
S, whereTdis a group with probabilistic convergence structure cd. Thus, if the triple ( T;S; ) is a prob-
abilistic convergence transformation group, then we can speak about the triple ( Td;S;) as probabilistic
convergence transformation group and may be called as discrete probabilistic convergence transformation
group .
Denition 5.9. The category of probabilistic convergence transformation groups denoted by PCON-
VTGRP consists of all triple ( T;S; ) of probabilistic convergence transformation groups as objects, and
all pair (f;g): (T;S; )!(T0;S0;0) as morphism, where
(TG1)Tf!T0is aPCONVGRP -morphism, i.e., a continuous group homomorphism;
(TG2)Sg!S0is aPCONV -morphism, i.e., a continuous mapping.
If (f;g): (T;S; )!(T0;S0;0) and (h;k): (T0;S0;0)!
T00;S00;00
arePCONVTGRP -morphisms,
then their composition is dened by: ( h;k)
(f;g) = (h
f;k
g), where (h
f;k
g): (T;S; )!
T00;S00;00
, whenceh
f:T!T00isPCONVGRP -morphism and h
g:S!S00isPCONV –
morphism.
Lemma 5.10. LetPCONV be the category of probabilistic convergence spaces and PCONVGRP be
the category of probabilistic convergence groups under the triangle function . IfPCONVTGRP denotes
the category of probabilistic convergence transformation groups, then we have the following covariant
functors
A:8
>><
>>:PCONVTGRP !PCONVGRP 
(T;
;t= (t')'2
+);(S;c= (c')'2
+);

7!
T;
;t

(f;g) 7! f

10 T. M. G. Ahsanullah, T. M. Baran, Fawzi Al-Thukair
and
B:8
>><
>>:PCONVTGRP !PCONV

(T;
;t= (t')'2
+);(S;c= (c')'2
+);

7! (S;c)
(f;g) 7!g
Now if we consider the category PCONVTGRP which is a subcategory of PCONV , then it follows
from [2] that for an object ( T;
;t) inPCONVGRP and object ( S;c) inPCONV , we have the product
probabilistic convergence structure, shortly denoted by TS, which is again an object in PCONV .
Furthermore, let us put C:=PCONVGRP PCONV denote the category, called the product category
ofPCONVGRP andPCONV (cf. [26], §II.3, pp. 36) having objects the pair ( T;S), whereT=

T;
;t

2jPCONVGRP j, andS= (S;c)2jPCONVj, and morphisms as the pair ( f;g), wherefis
a morphism in PCONVGRP which is a continuous group homomorphism while gis a morphism in
PCONV which is a continuous mapping; if ( f;g) and (f0;g0) are morphisms, then their composition is
dened as: ( f0;g0)
(f;g) := (f0f;g0g).
Lemma 5.11. Consider C:C!PCONVGRP andD:C!PCONV be the canonical projection
functors. Then there exists a unique covariant functor P:PCONVTGRP !C
PCONVTGRP
PCONVGRP  C PCONVABP
C D
such that CP=AandDP=B, where Pis given by
P:8
>><
>>:PCONVTGRP ! C

(T;
;t= (t')'2
+);(S;c= (c')'2
+);

7! (T;S)
(f;g) 7! (f;g)
Remark 5.12. The main issue in the preceding lemma is the construction of the unique mapping P.
However, it is important to note that the construction of Pis the only one possible, for if P((T;S; )) =
(T;S), then commutativity implies T=C
P((T;S; )) =A((T;S; )), the probabilistic convergence group
under the triangle function , andS=D
P((T;S; )) =B((T;S; )), the probabilistic convergence space.
Theorem 5.13. Let(Tj;Sj;j)j2Jbe a family of probabilistic convergence transformation groups. Then
the product in PCONVTGRP of its objects is the probabilistic convergence transformation groupsQ
jTj;Q
jSj;
together with the projections
(pi;qi):Q
jTj;Q
jSj;
!(Ti;Si;i)
whereQ
jTjandQ
jSjare the usual product in the categories PCONVGRP andPCONV , andpi,
qiare the usual projections. Moreover, is dened by ((tj)j;(sj)j) = (j(tj;sj))j.

Monad and its Eilenberg-Moore category 11
6 Example: Monad in the category C
Denition 6.1. Amonad on a category Xis a triple T= (T;e;m) consisting of an endofunctor T:X!
X, and natural transformations
e:idX!Tandm:TT!T
such that the diagrams below are commutative:
CeC//
f
T(C)
T(f)

DeD//T(D)T(T(C))mC//
T(T(f))
T(C)
T(f)

T(T(D))mD//T(D)
Also, it satises the commutativity of the following diagrams (multiplication law and left, right units
law):
T(T(T(C)))T(mC)! T(T(C))
??ymT(C)??ymC
T(T(C))mC! T(C)(a) (1)
T(C)eT(C)! T(T(C))T(eC) T(C)
idT(C)& mC#idT(C).
T(C)(b)
Denition 6.2. LetT= (T;e;m) be a monad on X. Then a T-algebra is a pair ( X;) consisting of an
objectXinXand a morphism TX!XinXsuch that the following diagrams commute:
XeX! TX
idX&#
X(a)
TTXT! TX
??ymX??y
TX!X(b) (2)
that is,
eX=idXand
T=
mX:T(T(X))!X.
Let (X;) and (Y;) beT-algebras, then a morphism f:X!YinXis called a morphism ofT-algebras,
from (X;) to (Y;), whenever the following diagram commutes:

12 T. M. G. Ahsanullah, T. M. Baran, Fawzi Al-Thukair
TX!X
??yTf??yf
TY!Y(c) (3)
Since composite in Xof morphisms of T-algebras are again morphisms of T-algebras, thus in this way we
obtain a category: the category ( XT;UT) of all T-algebras.
Theorem 6.3. LetC:=PCONVGRP PCONV . Dene the endofunctor T:C!CbyT(T;S) =
(T;TS)andT(f;g) = (f;fg),8(T;S)2jCj. Furthermore, e(T;S): (T;S)!(T;TS);(t;s)7!
(t;(e;s))withe(T;S)= (idT;eT
S), where eT
S:S!TS;is dened by eT
S(s) = (e;s), and m(T;S): (T;T(TS))!
(T;TS)withm(T;S)= (idT;mT
S)where mT
S:T(TS)!TSis dened by mT
S(t1;(t2;s)) = (t1t2;s),
t1;t22T;s2S. Then the triple T= (T;m;e)is a monad in C.
Proof. To show that the triple T= (T;m;e) is a monad, we need to check the following:
(a)eandmare well-dened and Tis a functor.
(b)e(T;S): (T;S)!T(T;S) and m(T;S):T(T(T;S))!T(T;S) are natural transformations, i.e., the
diagrams below commute:
(T;S)(idT;eT
S)//
(idT;idS)
(T;TS)
(idT;idTidS)

(T;S)
(idT;eT
S)//(T;TS) (b1)(T;T(TS))(idT;idT)//
(idT;mT
S)
(T;TS)
(idT;)

(T;TS)
(idT;)//(T;S) (b2)
(c) Finally, multiplication and unitary (left and right) laws are fullled, i.e.,
T(T(T(C)))T(mC)! T(T(C))
??ymT(C)??ymC
T(T(C))mC! T(C)(b3) (4)
and
T(C)eT(C)! T(T(C))T(eC) T(C)
idT(C)& mC#idT(C).
T(C)(b4)
are commutative.
We begin the proof one by one. Remark that we have

Monad and its Eilenberg-Moore category 13
e(T;S):idC(T;S)!T(T;S), (T;S)7!(T;TS), and
m(T;S):T(T(T;S))!T(T;S);(T;T(TS))7!(T;TS).
(a) Clearly, e(T;S)andm(T;S)are well-dened. Since the product TS2jPCONVj, for each (T;S)2
jCj, we have ( T;TS)2jCj. Secondly, for each ( f;g)2Mor (C),T((f;g)
(h;k)) =T(fh;gk ) =
(fh;fhgk) = (fh;(fg
hk)) = (f;fg)
(h;hk) =T(f;g)
T(h;k) and also, clearly T(id(T;S)) =
id(T;TS)=idT((T;S)), thus, Tis a functor. To show that eandmare natural transformation. In view
of [18] (cf. II, pp. 18), it follows at once that the diagram in (b1) is commutative; for (b2), we have:
(idT;)
(idT;idT) (t;(t1;(t2;s))) = (idT;) (t;(t1;(t2s))) = (idT;) (t;((t1t2);s)) = (t;(t1t2)s). On
the other side, we have 8t;t1;t22Tands2S: (idT;)

idT;mT
S

(t;(t1;(t2;s))) = (idT;) (t;(t1t2;s)) =
(t;(t1t2)s). So, (b2) is also commutative. Finally, we prove (c), i.e., multiplication, and left and right
unit laws hold. For, rst we note that the diagram below:
T(T(T(T;S)))Tm(T;S)! T(T(T;S))
??ymT(T;S)??ym(T;S)
T(T(T;S))m(T;S)! T(T;S)(5)
can be put into the following form:
(T;T(T(TS)))(idT;mT
TS)! (T;T(TS))
??y(idT;idTmT
S)??y(idT;mT
S)
(T;T(TS))(idT;mT
S)! (T;TS)(6)
whence we have to check that ( idT;mT
S)
(idT;idTmT
S) = (idT;mT
S)
(idT;mT
TS);
it suces to check the following:
mT
S
(idTmT
S)(t;(t1;(t2;s))) = mT
S[t;mT
S(t1;(t2;s))] = mT
S[t;(t1t2;s)]
= [t(t1t2);s] = [(tt1)t2;s] =mT
S
mT
TS(t;(t1;(t2;s)))
and
T(T;S)eT(T;S)! T(T(T;S))
??yTe(T;S)??ym(T;S)
T(T(T;S))m(T;S)! T(T;S)(7)
can be written as follows:

14 T. M. G. Ahsanullah, T. M. Baran, Fawzi Al-Thukair
(T;TS)(idT;eT
TS)! (T;T(TS))
??y(idT;idTeT
S)??y(idT;mT
S)
(T;T(TS))(idT;mT
S)! (T;TS)(8)
where we need to check that ( idT;mT
S)
(idT;idTeT
S) = (idT;mT
S)
(idT;eT
TS).
Here too it is enough to check the following, i.e.,
mT
S
(idTeT
S)(t;s) =mT
S[t;eT
S(s)] =mT
S[t;(e;s)] = (te;s) = (et;s)
=mT
S(e;(t;s)) =mT
S[eT
TS(t;s)] =mT
S
eT
TS(t;s)
This completes the proof that the triple T= (T;m;e) is a monad on C.
7 Eilenberg-Moore algebras
Denition 7.1. LetT= (T;m;e) be a monad. An Eilenberg-Moore algebra or aT-algebra ((T;S);(f;g))
is a pair consisting of an object ( T;S) inC(the underlying object of the algebra), and a morphism
(f;g):T((T;S)) = (T;TS)!(T;S) ofC(called the structure map of the algebra) which makes the
two diagrams below
idC(T;S)e(T;S)! T(T;S)
id(T;S)& (f;g)#
(T;S)(a)
and
TT(T;S)T(f;g)=(f;fg)! T(T;S)
??y(f;m(T;S))??y(f;g)
T(T;S)(f;g)! (T;S) ( b)(9)
commute (the rst diagram (a) is unit law, and the second diagram (b) is associative law).
A morphism ( h;k): ((T;S);(f;g))!((T0;S0);(f0;g0)) ofT-algebra is a morphism
(h;k): (T;S)!(T0;S0) ofCsuch that the diagram
T(T;S)(f;g)! (T;S)
??yT(h;k)??y(h;k)
T(T0;S0)(f0;g0)! (T0;S0)(10)
commute, i.e., ( f0;g0)
T(h;k) = (h;k)
(f;g).
The category CTofT-algebras andT-morphisms is denoted by ALG T, also called Eilenberg-Moore cate-
gory of algebras .

Monad and its Eilenberg-Moore category 15
Lemma 7.2. LetT= (T;e;m)be a monad in C. Then the pair ((T;S);(idT;))is aT-algebra of the
monad T, where:TS!Sthe action of probabilistic convergence group on probabilistic convergence
space.
Proof. We need to prove that the following diagrams
(T;S) =idC(T;S)(idT;eS
T)! T(T;S) = (T;TS)
(idT;idS)& (idT;)#
(T;S)(a)
and
T(T(T;S)) = (T;T(TS))(idT;idT)! T(T;S) = (T;TS)
??y(idT;mT
S)??y(idT;)
T(T;S) = (T;TS)(idT;)! (T;S)(b) (11)
are commutative, where ( T;S)2jCjand (idT;) is a morphism between T(T;S) and (T;S).
To prove the commutativity in (a), we only check 
eT
S=idS, i.e.,

eT
S(s) =(e;s) =es=s=idS(s)
for alls2S; for (b) we proceed as follows:

(idT)(t;(t1;s)) =[t;(t1;s)] =t(t1s) = (tt1)s=(tt1;s)

(mT
S(t;(t1;s))) =
mT
S(t;(t1;s))
for allt;t12Tands2S, i.e., we get 
(idT) =
mT
S.
Theorem 7.3. The concrete category (PCONVTGRP ;P)overCis isomorphic to (CT;HT)overC
where CTis the category of T-algebras associated with the monad T= (T;m;e). Furthermore, for the
forgetful functor HT:CT!C, the following diagram of functors commutes:
PCONVTGRP CT
CP0
P HT
that is, HTP0=P.
Proof. Consider P0:PCONVTGRP !CT. To show that P0is an isomorphism, it suces to check
that it is indeed a functor in the present situation, and that this functor is full, faithful and, bijective on
objects, i.e., P0:jPCONVTGRP j!j CTjis bijective.
First note that P0:PCONVTGRP !CT, (T;S; )7!((T;S);(idT;)) is a functor. In fact, for
(f;g);(h;k)2Mor(PCONVTGRP ), we have P0(h;k)
P0(f;g) = (h;k)
(f;g) = (hf;kg ) =P0(hf;kg );

16 T. M. G. Ahsanullah, T. M. Baran, Fawzi Al-Thukair
also,P0(idT;idS) = (idT;idS) = (id;id )(T;S)=idP0(T;S).
To check P0is faithful, let P0(f;g) =P0(f0;g0). Thenf=f0andg=g0. Furthermore, let ( f;fg)2
Mor(CT). Then there exists ( h;k)2Mor(PCONVTGRP ) such that P0(h;k) = (f;fg), by taking h=
fwhich is a continuous group-homomorphism between probabilistic convergence groups, while k=fg,
a continuous mapping between probabilistic convergence spaces, we are done. So, P0is full (epic). For, P0
to be faithful (objectwise), let P0(T;S; ) =P0(T0;S0;0), for (T;S; );(T0;S0;0)2jPCONVTGRP j.
Then P0(T;S; ) = ((T;S);(eT;))2jCTj, and P0(T0;S0;0) = ((T0;S0);(eT0;0))2jCTj. Then [T=T0,
S=S0,idT=idT0; (eT;s) =0(eT0;s0)] implying that ( T;S; ) = (T0;S0;0):
Now pick (( T;S);(idT;)))2jCTj. There exists ( T;S;0) = ((T;S;0= (eT;))2jPCONVTGRP j
such that P0(T;S;0) = ((T;S);(eT;)), where T(T;S) = (T;TS)0=(eT;)! (T;S) is a morphism in C,
and:TS!Sis an action of TonS, because of the fact that the diagram in the denition of
T-algebra need to be commutative (cf. Denition 5.1, and Lemma 5.5), i.e., a part of the diagram (cf.
equation (10)) is to satisfy: 
eT
S=idTand
mT
S=
(idT). This ends the proof that P0is an
isomorphism, i.e., it is full, faithful and bijective on objects.
To prove the nal part, note that by denition of P0, it a bijection on objects between
PCONVTGRP andCT, while HTis a forgetful functor. Thus, we have for any ( T;S; )2jPCONTGRPj:
HTP0((T;S; )) =HT((T;S);(idT;)) = (T;S) =P(T;S; );i.e.,HTP0=P.
Corollary 7.4. The functor P:PCONVTGRP !Chas a left adjoint.
Proof. Due to the preceding Theorem 7.3, we have CTis isomorphic to PCONVTGRP by symmetry
given the fact that isomorphism to is an equivalence relation. Furthermore, GTis the left adjoint of HT.
CF! PCONVTGRP
GT& P#
CT
such that PF=GT
Theorem 7.5. Consider the adjunction (; ;e;") :PMETGRP!PMPCONVGRP , where
PMETGRP PMPCONVGRP

are the functors. Specically, the two functors :PMETGRP!PMPCONVGRP and
:PMPCONVGRP !PMETGRP have composite  = T:PMETGRP!PMETGRP an
endofunctor. The unit of the adjunction is the natural transformation e:idPMETGRP!Tand the
counit":  !idPMPCONGRP of the adjunction yields a natural transformation m= " :   =
TT!  = T. Then ( ;e; ")is a monad in PMETGRP .
Proof. We need to show that T:PMETGRP!PMETGRP dened by ( S;
;F) = (S;
;Fc)
and (f) =fis a functor and e:idPMETGRP!T=  (the unit) and m= ":TT!T

Monad and its Eilenberg-Moore category 17
(multiplication) are natural transformations satisfying associative law and (left and right) unit law: we
have the commutativity of the following diagrams:
TTT =   Tm=  "//
mT= " 
TT=  
m= "

TT=  m= "//T= (a)    "//
"

"

 "//idPMPCONVGRP(b)
e !   e 
id & "#id .
(c)
First, we check that the composition : PMETGRP!PMETGRP of functors and  is again
a functor. In fact, for any f;g2Mor(PMETGRP ), wheref:S!S0andg:S0!Tare non-expansive,
thengf:S!Tis non-expansive and is a morphism in PMETGRP . Now we have
(fg) = ((f)(g)) = (fg) = (f) (g) = ( ((f)))( ((g))) = ( (f))( (g));
and clearly, ( idS) =id (S).
In view of the Theorem 4.11, we obtain the left and right units (diagram (c)):
( ")
(e )(S;
;F) = ( ")[e(S;
;Fc)] = ( ")(S;
;Fc) = (S;
;Fc) =idPMETGRP (S;
;F);
( ")
( e)(S;
;F) = ( ")
(S;
;F) = ( ")(S;
;Fc) = (S;
Fc) =idPMETGRP (S;
;F):
From diagram (b), we have
"
(" )(S;
;c) ="(S;
;c) = (S;
;c) =idPMPCONVGRP (S;
;c)
"
(" )(S;
;c) ="( (S;
;c) ="(S;
;c) =idPMPCONVGRP (S;
;c)
proving the commutativity of the diagram in question. For the diagram (a), i.e., to show m
mT=m
Tm,
we apply Theorem 4.11:
( ")
( " )(S;
;F) = ( ")[ "(S;
;Fc)] = "( "(S;
;cFc=c))
= ( ")[ (S;
;c)] = ( ")(S;
;Fc) = (S;
;Fc)
Similarly, on the other hand, we get
( ")
[ ")(S;
;F))] = "(S;
;Fc) = (S;
;Fc)
This proves the commutativity of the diagram (a), i.e., associative law holds.
Therefore, (  ;e; ") is indeed a monad in PMETGRP .

18 T. M. G. Ahsanullah, T. M. Baran, Fawzi Al-Thukair
Theorem 7.6. LetT= (T;e;m)be a monad in C=PCONVGRP PCONV , then the set of all
T-algebras and their morphisms form a category CT. There is an adjunction

GT;HT;eT;"T

:C*CT
in which the functors GT, and HT:
CTCHT
GT
are given as follows:
HT:CT!C;((T;S);(idT;))!(T;S)
which is a forgetful functor, forgetting the structure map of each T-algebra, and
GT:C!CT;(T;S)!
T(T;S);m(T;S)

=
(T;TS);(idT;mT
S)

while eT=e: (T;S)!T(T;S) = (T;TS)and"T((T;S);(idT;)) = (idT;).
This adjunction gives rise to a monad
HTGT;eT;HT"TGT

which coincides with the original monad
T= (T;e;m).
Proof. Note that if ( h;k): ((T;S);(f;g))!((T0;S0);(f0;g0)) and
(h0;k0): ((T0;S0);(f0;g0))!
(T00;S00);(f00;g00)
are morphisms of T-algebras, so is their composite
(h0;k0)
(h;k); with this composition, clearly T-algebras form a category CT, and HT:CT!Cis a
functor.
Now we show that GT:C!CT;(T;S)!
T(T;S);m(T;S)

=
(T;TS);(idT;mT
S)

is a functor.
Note that that for each ( T;S)2 jCj, the pair
T(T;S);m(T;S)

= ((T;TS);(idT;(idT))) is a
T-algebra (called freeT-algebra generated by ( T;S)), making the diagram (b) in Denition 7.1 is com-
mutative; similarly, (a) in the Denition 7.1 shows the unit law holds. Hence HT:C!CTgives rise to
a functor. Looking at the composition of functors, we have
HTGT(T;S) =HT
GT(T;S)

=HT
T(T;S);m(T;S)

=T(T;S).
Thus, it follows immediately that the unit e:idC!T(T;S) of the given monad Tis a natural transfor-
mation
e=eT:idCHTGT.
Next looking at other way round, we have
GTHT((T;S);(idT;)) =GT
HT((T;S);(idT;))

=GT(T;S) =
T(T;S);m(T;S)

.
From Denition 6.2 of T-algebra ((T;S);(f;g) = (idT;)), and gure (b) we know that the structure map
(f;g) = (idT;):T(T;S)!(T;S), i.e., (idT;): (T;TS)!(T;S) is a morphism
T(T;S);m(T;S)

!
((T;S);(idT;)) ofT-algebras. Thus by denition of morphism of T-algebras,

Monad and its Eilenberg-Moore category 19
"T= (f;g)(= (idT;)):GTHT(T;S)!((T;S);(idT;))
is a natural transformation.
The triangular identities for an adjunction is given by
T(T;S) = (T;TS)T(e(T;S))=T(idT;eT
S)=(idT;(idTeT
S))! T2(T;S) = (T;T(TS))
idT(T;S)& m(T;S)=(idT;idT)#
T(T;S) = (T;TS):
This hold good due to the unit law. Indeed, for any t2T, and (t;s)2TSin conjunction with the
mappings: e(T;S): (T;S)!(T;TS), whence e(T;S):= (idT;eT
S), and also, T(e(T;S)) =T(idT;eT
S) =
(idT;idTeT
S), where eT
S:S!TS;s7!(e;s), we have
(idT;idT)

idT;(idTeT
S)

(t;(t;s)) = (idT;idT) (t;(t;(e;s)))
= (t;(t;es)) = (idT;idTidS)(t;(t;s)) =idT(T;S)(t;(t;s)) =id(T;S);
and
(T;S)e(T;S)=(idT;eT
S)! T(T;S) = (T;TS)
id(T;S)=(idT;idS)& (idT;)#
(T;S)
for which the above diagram, we have for any t2Tands2S:
(idT;)
e(T;S)= (idT;)
(idT;eT
S)(t;s) = (idT;)(t;(e;s)) = (t;es) = (t;s) = (idT;idS)(t;s)
Therefore, eTand"Tdene an adjunction. This adjunction gives rise to a monad
HTGT;eT;HT"TGT

,
the endofunctor HTGTwhich yields original T, and its unit is the original unit e, and the multiplication
mT=HT"TGT, for all (T;S) inCis given by:
mT(T;S) =HT"TGT(T;S) =HT"T
T(T;S);m(T;S)

=HT(m(T;S)) =m(T;S)
which produces the original multiplication mofT, i.e., one obtains the original monad T= (T;e;m).
This ends the proof.
Theorem 7.7. The concrete category (PCONVTGRP ;P)over the category Cis monadic.
Proof. This follows from the fact that it has free objects and the associated comparison functor ( PCONVTGRP ;P)P0!
(CT;HT) is isomorphic.
Theorem 7.8. Every Eilenberg-Moore category of a monad T= (T;m;e)is closed under the formation
of mono-sources in ALG T.
Proof. This is an immediate consequence of the Lemma 20.11 [1].

20 T. M. G. Ahsanullah, T. M. Baran, Fawzi Al-Thukair
8 Kleisli category with respect to a subcategory of the category of proba-
bilistic convergence spaces
The objects of the Kleisli category CTassociated with the monad T= (T;e;m)
onC:=PCONVGRPPCONV are the same objects ( T;S) ofC, whereTis the probabilistic conver-
gence group while Sis the probabilistic convergence space. But a morphisms ( f;g): (T;S)*(T0;S0)
inCTis simply a C-morphism ( f;g): (T;S)!T(T0;S0) from (T;S) to the extension T(T;0;S0) of
(T0;S0), i.e., (f;g): (T;S)!T(T0;S0) = (T0;T0S0). The Kleisli composition , we denote it by ,
of (f;g): (T;S)*(T0;S0) and (f0;g0): (T0;S0)*(T00;S00) inCTis dened via the composition
inCas
follows:
(f0;g0)(f;g) =m(T00;S00)
T(f0;g0)
(f;g)
which can be viewed in the following arrows:
(T;S)(f0;g0)(f;g)! T(T00;S00) = (T;S)(f;g)!T(T0;S0)T(f0;g0)!T(T(T00;S00))m(T00;S00)! T(T00;S00),
The identity id(T;S): (T;S)*(T;S) in this category is just the component
e(T;S): (T;S)!T(T;S) = (T;TS) of the unit e.
Lemma 8.1. LetT= (T;e;m)be a monad in C. Then (CT;eT;)gives rise to a Kleisli category.
Proof. According to the given condition, as Tis a monad on C,m:T
T!Tis a natural transformation,
then for any C-morphism ( T;S)!T(T;S), we have the commutativity of the following diagram:
T(T(T00;S00))T(T(f00;g00))! T(T(T(T000;S000)))
??ym(T00;S00)??ymT(T00;S00)
T(T00;S00)T(f00;g00)! T(T(T00;S00))(12)
that is
T(f00;g00)
m(T00;S00)=mT(T000;S000)
T(T(f00;g00)),
which produces the following diagram in our sense:
(T;T(TS))T(T(f00;g00))=(idT;idT(idTeT
S))! (T;T(T(TS)))
??ym(T;S)=(idT;mT
S)=(idT;idT)??ym(T;TS)=(idT;idTmT
S)
(T;TS)T(f00;g00)=(idT;idTeT
S)! (T;T(TS))(13)
that is, the following holds:
(idT;idTeT
S)
(idT;idT) = (idT;idTmT
S)
(idT;idT(idTeT
S))

Monad and its Eilenberg-Moore category 21
In fact, for any t;u2Tands2S, it suces to check the following:
(idTmT
S)
(idT(idTeT
S))(t;(u;s)) = (idTmT
S)(t;(u;(e;s))) = (t;mT
S(u;(e;s)) = (t;(u;s))
while
(idTeT
S)
(idT)(t;(u;s)) = (idTeT
S)(t;(u;s)) = (t;eT
S
(u;s)) = (t;idTS(u;s)) = (t;(u;s)):
One can check that the associative law holds good under the Kleisli composition , while the remaining
conditions can be seen as follows. But rst note that from the given monad T, we have e:idC!T
is a natural transformation. So, for any C-morphism ( f;g): (T;S)!(T;TS), which in our case as
mentioned before, we take ( f;g) = (idT;eT
S), then we have for any t2Tands2S:
[e(T;S)(idT;eT
S)] = [( m(T;S)
T(e(T;S)))
(idT;eT
S):
Note that ( m(T;S)
T(e(T;S))) =idT(T;S);where e(T;S): (T;S)!T(T;S).
In fact,
T(T;S) = (T;TS)idT(T;S)!T(T;S) = (T;TS)
=

T(T;S) = (T;TS)T(e(T;S))! T(T(T;S)) = (T;T(TS))m(T;S)!T(T;S) = (T;TS)
:
Thus, we have
[(m(T;S)
(T(e(T;S)))
(idT;eT
S))] =idT(T;S)
(idS;eT
S) = (idT;eT
S)
which follows from the following arrows:
(T;S)e(T;S)=(idT;eT
S)! (T;TS) =T(T;S)idT(T;S)!T(T;S) =
(T;S)(idT;eT
S)! T(T;S)
that is, e(T;S)(idT;eT
S) = (idT;eT
S).
Therefore, we obtained Kleisli category CT= (T;e;), where the object function is given by T:jCj!j Cj,
the composition is , and the identity is e.
Lemma 8.2. A Kleisli category has a forgetful functor DT:CT!Cas described below:
DT:8
>><
>>:CT! C
(T;S)7! (T;TS)
((f;g): (T;S)*(T0;S0))7! (m(T0;S0)
T(f;g):T(T;S)!T(T0;S0))
This functor DThas a left adjoint; in fact, a left embedding as described below:

22 T. M. G. Ahsanullah, T. M. Baran, Fawzi Al-Thukair
Lemma 8.3.
ET:8
>><
>>:C! CT
(T;S)7! (T;S)
((f;g): (T;S)*(T0;S0))7! (e(T0;S0)
(f;g): (T;S)*(T0;S0)
The unit eT:idC!DTET=Tof this adjunction is eand the components of the counit "T:ETDT!
idCTare simply the morphisms idT(T;S):T(T;S)!T(T;S), i.e.,id(T;TS): (T;TS)!(T;TS)
inC.
We can put our preceding paragraph into the following
Theorem 8.4. The monad (DTET;eT;"T)associated with the adjunction (DT;ET;eT;"T)gives rise to
the original monad T= (T;e;m).
9 conclusion
Considering the product category C=PCONVGRP PCONV of the categories of probabilistic con-
vergence groups under triangle function , and probabilistic convergence spaces – a subcategory of
PCONV , we have constructed a monad T= (T;e;m) on Cand the corresponding category of alge-
bras, so-called Eilenberg-Moore category of algebras, CT. We showed that this category CTis isomorphic
to the category of probabilistic convergence transformation groups, PCONVTGRP – the category that
we have introduced earlier in [3]. Exploiting the Theorems 6.3, 7.3, 7.5 and, particularly, Theorem 7.6,
one can at ease obtain the dual of these results: comonad and coalgebra , and hence the other results
thereof are achievable. Furthermore, we have touched upon a very few results on Kleisli category in
relation to the category of probabilistic convergence spaces.
Since both of the categories: PCONVGRP andPCONV are topological categories, one can easily
obtain various results on limits and colimits ; however, due to space constraint we refrain here in this
manuscript to include those results. We intend to look into various subcategories of the category of
probabilistic convergence transformation groups in our future papers.
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Corresponding author:1;T M G Ahsanullah,
Email: tmga1@ksu.edu.sa
2Fawzi Al-Thukair
1;2Department of Mathematics,
King Saud University,
College of Science, Riyadh 11451, Saudi Arabia.
Email: thukair@ksu.edu.sa
yTesnim Meryem Baran
MEB, Kayseri,
Turkey,
Email: mor.takunya@gmail.com