IDEALS IN CA-AG-GROUPOIDS M. Iqbal and I. Ahmad
Department of Mathematics, University of Malakand, Chakdara,
Khyber Pakhtunkhwa, Pakistan
e-mails: [email protected], [email protected]
(Received 16 October 2016; accepted 20 July 2017)
An AG-groupoidS satisfying the identity u(vw) = w(uv)∀u, v, w ∈ S is called a CA-AG-groupoid [1]. This article is devoted to the study of various characterizations of (left/right) ideals in CA-AG-groupoids and to the relationships between (left/right) connected sets and (left/right) ideals in CA-AG-groupoids.
Key words : AG-groupoid; LA-semigroup; CA-AG-groupoid; ideals; connected sets.
1. INTRODUCTION ANDPRELIMINARIES
A groupoid(G,·) or simplyGsatisfy(uv)w = (wv)u∀u, v,w ∈ G(called left invertive law [2])
is called an Abel-Grassmann’s groupoid (in short AG-groupoid [3]). The said structure is called
upon by different names by different authors, such as left almost semigroup (LA-semigroup) [2],
right modular groupoid [4], left invertive groupoid [5]. Throughout the article we will denote an
AG-groupoid simply bySotherwise stated else. Also to avoid excessive parenthesization and dots, we will
useuvforu·v,uv·wtfor(uv)(wt), and(uv·w)tfor((uv)w)t. The medial law:uv·wt=uw·vt
always holds inS [6, Lemma 1.1(i)]. Left identity may or may not contained inS; however, if S
contains left identity then it is unique [7] andS with left identity always satisfies the paramedial
law: uv ·wt = tv ·wu[6, Lemma 1.2(ii)]. Now, we define some elementary aspects and quote
it is simultaneously right and left alternative [9]. An elemente ∈ S is called left identity (right
identity) of S if eu = u∀u ∈ S (ue = u∀u ∈ S). In S left identity does not implies right
identity, however a right identity is always a left identity [7, Theorem 2.3]. S is called AG* [10],
if∀u, v, w ∈ S, (uv)w = v(uw) and is called cyclic associative AG-groupoid (in short
CA-AG-groupoid) ifu(vw) = w(uv) [1]. In [1] Iqbal et al. enumerated CA-AG-groupoids up to order6
and further classified it into different subclasses. Further, they introduced CA-test for verification
of arbitrary AG-groupoid to be cyclic associative, and studied some fundamental properties of
CA-AG-groupoids. The same authors in [11] discussed different aspect of cancellativity of an element in
CA-AG-groupoid and provided a partial solution to an open problem given in [12]. For detail study
of properties of CA-AG-groupoids we recommend [1, 11].
For any two subsets (by a subset in the article we means a non-empty subset) M, N ofS and u∈S,uM ={um: m∈M},M u={mu: m∈M}andM N ={mn: m∈M∧n∈N}. IfS
is a CA-AG-groupoid andL, M, N ⊆S, then for alll∈L,m∈M andn∈N, l(mn) =n(lm) =
m(nl), which implies thatL(M N) =N(LM) =M(N L).
An AG-subgroupoid ofS is a subsetH ofS such thatH2 ⊆ H. A subsetK ofS is said to be right (left) ideal ifKS ⊆K(SK ⊆K)andK is called a two-sided ideal (in short an ideal) ifK is
simultaneously right and left ideal ofS [13]. L, M ⊆ Sare called right connected (left connected)
sets ifLS ⊆M andM S⊆L(SL ⊆M andSM ⊆L). LandM are called connected sets if they
are simultaneously right and left connected [13].
It is proved in [14] that if an AG-groupoidScontains left identityethenSS =SandS=eS= Se, i.e. egenerates the same left and right ideals. Also ifShas left identity then every right idealR
is ideal,SRis left ideal andRSis right ideal ofS. Also for left idealL, right idealRanda∈S,aL
is left ideal,R2 is an ideal ofS. The set of ideals of a regular LA-semigroup form a semilattice, for each ideal there exists a minimal prime ideal.
In [15] Mushtaq et al. proved that in AG-3-bandS(i)right ideal implies left ideal and vice versa,
(ii) for two idealsM andN ofS,(a) M N is an ideal and (b) M N andN M are connected sets.
They also proved that if the set of all ideals ofSis totally order then every ideal ofSis prime and the
set of ideals ofSis a semilattice.
2. IDEAL INCA-AG-GROUPOIDS
To begin with, we illustrate the existence of (left/right) ideals in CA-AG-groupoid by providing
for a CA-AG-groupoid. We also provide examples of subsets of CA-AG-groupoid which are right as
well as left ideal, and verify that a subset ofSmay or may not be an ideal. Furthermore, by examples
we demonstrate that CA-AG-groupoid may have no ideal, one ideal or more than one ideals.
Example 1 : LetS ={0,1,2,3,4,5,6,7}and the binary operation onSis defined by the
follow-ing Cayley’s table:
· 0 1 2 3 4 5 6 7
0 3 3 5 3 3 3 7 3
1 4 3 3 3 3 7 3 3
2 3 6 3 3 7 3 3 3
3 3 3 3 3 3 3 3 3
4 3 3 3 3 3 3 3 3
5 3 3 3 3 3 3 3 3
6 3 3 3 3 3 3 3 3
7 3 3 3 3 3 3 3 3
ThenS is a CA-AG-groupoid (can be verified by CA-test [1]). Furthermore, we have the following
observations;
(i) ForL ={0,3,4,7}, SL = {3,4,7} ⊆ L, thusLis left ideal of S. However,Lis not right
ideal asLS ={3,5,7}*L. Similarly,{2,3,5,7}and{0,2,3,4,5,7}are left ideals but not
right ideals.
(ii) Each of the subsets {3,4},{3,5},{3,6},{3,4,5},{3,4,6},{3,5,6},{0,3,5,7},{3,4,5,6}, {1,3,4,7}is a right ideal but not a left ideal ofS.
(iii) {3},{3,7},{3,4,7},{3,5,6},{3,5,7},{3,6,7}, {3,4,5,7}, {3,4,6,7}, {3,4,5,6,7}, {3,4,5,7},{3,4,6,7},{3,5,6,7}, etc. are ideals ofS.
(iv) {0,1},{0,2},{0,3},{1,2},{1,4},{1,5},{1,6},{1,7},{2,5},{2,6},{2,7},{4,5},{4,6}, {4,7},{5,6},{5,7},{6,7},{0,1,6},{0,1,7},{1,2,3},{1,3,4},{1,3,5},{1,3,6},{0,1,4}, {0,1,5},{0,2,4}, {0,2,5}, {2,3,7},{4,5,6}, {4,5,7},{4,6,7},{5,6,7},{0,1,2,4}, {0,1,2,5},{4,5,6,7}, etc are neither right nor left ideals ofS.
(v) TakingM1 = {3,4,5,6}andN1 = {3,4,7}, then M1 andN1 are right connected sets, as: M1S = {3} ⊆ N1 andN1S = {3} ⊆ M1. However,M1 andN1 are not left connected, as: SN1 ={3,7} *M1. HenceM1 andN1 are non-connected sets. Note thatM1 is right ideal
(vi) TakingM2={2,3,7}andN2 ={3,5}, thenSM2 ={3,5} ⊆N2andSN2 ={3,7} ⊆M2, thus M2 and N2 are left connected sets. However,M2 and N2 are not right connected, as: M2S = {3,6,7} *N2. Note thatM2 is not an ideal andN2 is right ideal ofS, but stillM2 andN2 are left connected sets, however both are not right connected.
Noteworthy that if right identity contains in an AG-groupoid then it form a commutative
semi-group [7], thus right and left ideals coincide in such case. Also, as in CA-AG-semi-groupoid left identity
implies (right) identity [11], thus if left identity contains in CA-AG-groupoid then right and left ideals
coincide.
Example 2 : LetZnbe the CA-AG-groupoid as mentioned, then under usual multiplication
mod-ulon.
(i) Z4 ={0,1,2,3}is CA-AG-groupoid,{0,2}is the only ideal ofZ4.
(ii) Z8 ={0,1,2,3,4,5,6,7}is CA-AG-groupoid. {0,2,4,6}and{0,4}are ideals ofZ8. How-ever,{2,4,6},{0,3}and{0,6}are not ideals ofZ8. Hence, a subset of CA-AG-groupoid may not be an ideal.
(iii) Z12 = {0,1, ...,11} is CA-AG-groupoid. {0,6}, {0,4,8}, {0,2,4,6,8,10} and{0,3,6,9} are ideals ofZ12.
Example 3 :Z+∪ {0}is CA-AG-groupoid andk(Z+∪ {0}) ={0, k,2k,3k, ...}is an ideal of Z+∪ {0}for every integerk∈Z+.
Example 4 :
(i) D2 = ("
u 0 0 v
#
|u, v∈Z )
is an infinite CA-AG-groupoid under multiplication andA2 = ("
u 0 0 u
#
|u∈Z )
, is CA-AG-subgroupoid of D2. D2 have no proper ideal. Similarly,
D3 =
u 0 0 0 v 0 0 0 w
|u, v, w ∈Z
is CA-AG-groupoid under multiplication having no
proper ideal. Generally Dn, set of all diagonal matrices over Z, is CA-AG-groupoid under
multiplication having no proper ideal.
(ii) Gn ={(aij)|aij ∈Z}, ofn×nmatrices is CA-AG-groupoid under usual addition, forn=
1,2,3.., n. Gnhas no proper ideal in their respective order. Note thatUn ={set of all upper
2.1 Characterization of CA-AG-groupoids by Ideals
In the following, we establish relationship between (left/right) ideals and (left/right) connected sets
in a CA-AG-groupoidS. We prove that for right idealR, left idealLandM ⊆S,M R,RS,SRare
left ideals,LM is left ideal ifLM =M L. We also prove that for right idealsR1, R2and left ideals L1, L2 ofS, R1R2 andR2R1 are left ideals, furthermoreL1L2 andL2L1 are connected sets. We depict these results by giving some sufficient examples. Our starting points is the following lemma.
Lemma 1 — LetRbe a right ideal of CA-AG-groupoidS andM ⊆S, thenM R, RSandSR
are left ideals.
PROOF: By cyclic associativity
S(M R) =R(SM) =M(RS)⊆M R⇒S(M R)⊆M R,
and S(RS) =S(SR) =R(SS)⊆RS ⇒S(RS)⊆RS,
also S(SR) =R(SS) =S(RS)⊆SR ⇒S(SR)⊆SR.
Hence, the result follows. 2
In the following, we provide illustrative example for Lemma 1, and also provide a counterexample
to verify that if right idealRis replaced by left idealLin Lemma 1 thenM Lmay not be a left ideal.
Example 5 : Recall example 1,R = {1,3,4,7}andL = {2,3,5,7}are respectively right and
left ideal ofS. TakingM ={0,5} ⊆S, thenM R ={3}andS(M R) = {3} ⊆ M R, thusM Ris
left ideal. Now,M L={3,5}andS(M L) ={3,7}*M L, thusM Lis not a left ideal.
Theorem 1 — (i) IfR1 andR2 are right ideals of a CA-AG-groupoidS, thenR1R2 andR2R1
are left ideals,
(ii) IfL1andL2are left ideals of CA-AG-groupoidS, thenL1L2andL2L1 are connected sets.
PROOF: (i) By cyclic associativity
S(R1R2) =R2(SR1) =R1(R2S)⊆R1R2⇒S(R1R2)⊆R1R2.
Hence,R1R2is a left ideal ofS. Similarly,S(R2R1)⊆R2R1, thusR2R1 is also a left ideal.
(ii) By cyclic associativity
S(L1L2) =L2(SL1)⊆L2L1andS(L2L1) =L1(SL2)⊆L1L2.
Hence,L1L2andL2L1are left connected sets. Now, by left invertive law
Thus,L1L2andL2L1are also right connected. 2
We illustrate Theorem 1, by the following example.
Example 6 : Recall Example 1,Sis a CA-AG-groupoid.
(i) R1 ={3,4,5}andR2 ={1,3,4,7}are right ideals ofS. NowR1R2 ={3},R2R1={3,7}, S(R1R2) ={3} ⊆R1R2andS(R2R1) ={3} ⊆R2R1, thusR1R2 andR2R1are left ideals ofS.
(ii) TakingL1 = {0,2,3,4,5,7}andL2 = {2,3,5,7}thenL1andL2 are left ideals ofS. Now L1L2 = {3,5},L2L1 = {3,7},S(L1L2) = {3,7} ⊆ L2L1 andS(L2L1) = {3} ⊆ L1L2, thus L1L2 andL2L1 are left connected. Again, (L1L2)S = {3} ⊆ L2L1 and (L2L1)S = {3} ⊆ L1L2, thus L1L2 and L2L1 are right connected, too. Hence, L1L2 and L2L1 are connected sets.
Coupling part(i)and part(ii)of Theorem 1, we have the following corollary.
Corollary 1 — IfR1,R2are right ideals of CA-AG-groupoid, thenR1R2·R2R1andR2R1·R1R2 are connected sets.
Example 7 : As shown in Example 6 part(i), for right idealsR1={3,4,5}andR2={1,3,4,7}, R1R2 = {3}andR2R1 ={3,7}are left ideals. NowR1R2·R2R1 ={3},R2R1·R1R2 = {3}, alsoS(R1R2·R2R1) = {3} ⊆ R2R1·R1R2 andS(R2R1 ·R1R2) = {3} ⊆R1R2·R2R1, thus R1R2·R2R1andR2R1·R1R2are left connected. Again,(R1R2·R2R1)S={3} ⊆R2R1·R1R2
and(R2R1·R1R2)S ={3} ⊆R1R2·R2R1, thusR1R2·R2R1andR2R1·R1R2are right connected, too. Hence,R1R2·R2R1 andR2R1·R1R2are connected sets.
Lemma 2 — If R1 andR2 are right ideals of CA-AG*-groupoidS, then R1R2 andR2R1 are ideals ofS.
PROOF: As proved in Theorem 1 part(i)that for right idealsR1 andR2,R1R2 andR2R1 are left ideals. Now, we proceed to show thatR1R2andR2R1are right ideals. By left invertive law, AG* and cyclic associativity
(R1R2)S = (SR2)R1=R2(SR1) =R1(R2S)⊆R1R2⇒(R1R2)S ⊆R1R2.
Similarly,(R2R1)S⊆R2R1. Hence, the result follows. 2
PROOF: By AG* and left invertive law
S(R1R2) = (R1S)R2 = (R2S)R1 ⊆R2R1.
Similarly,S(R2R1)⊆R1R2, thusR1R2 andR2R1are left connected. Again
(R1R2)S=R2(R1S)⊆R2R1.
Similarly,(R2R1)S ⊆R1R2, thusR1R2 andR2R1are right connected. 2
Lemma 4 — IfRandLare right and left ideals of CA-AG*-groupoidS, then
(i) RLandLRare ideals ofS,
(ii) RLandLRare connected sets.
PROOF: (i) By using left invertive law, AG* and cyclic associativity
(LR)S= (SR)L=R(SL) =L(RS)⊆LR,
and S(LR) =R(SL) =L(RS) = (RL)S = (SL)R⊆LR.
Thus,LRis an ideal. Again
(RL)S=L(RS) =S(LR) =R(SL)⊆RL,
and S(RL) =L(SR) =R(LS) =S(RL) = (RS)L⊆RL.
Hence,RLis an ideal.
(ii) By AG* and cyclic associativity
S(LR) =R(SL)⊆RSandS(RL) =L(SR) = (SL)R⊆LR,
thusLRandRLare left connected. Again by left invertive law and AG*
(LR)S= (SR)L=R(SL)⊆RS and (RL)S= (SL)R⊆LR,
thusLRandRLare left connected. 2
In the following, we prove that for left ideal of CA-AG-groupoidSandM ⊆S,LMis left ideal
ifLM =M L, we also prove thatLS andSLare left ideals and connected sets, and generalize this
result and prove that((((LS)S)S)...)SandS(...(S(S(SL))))are left ideals and connected sets. We
further prove that if the conditionLM =M Lis modify toM L⊆LM then in such situationLM is
Lemma 5 — LetLis left ideal of CA-AG-groupoidSandM ⊆S. Then
(i) LM is left ideal iffLM =M L,
(ii) LS andSLare left ideals,
(iii) LSandSLare connected sets.
PROOF:
(i) LetLM =M L, by cyclic associativity
S(LM) =M(SL)⊆M L=LM ⇒S(LM)⊆LM.
Hence, LM is left ideal ofS. Conversely, letLM is left ideal ofS. Assume contrary that LM 6=M L. ThenS(LM) =M(SL)⊆M L6=LM, impliesLMis not a left ideal, which is
contradiction. Hence,LM =M L.
(ii) By cyclic associativity
S(LS) =S(SL) =L(SS)⊆LS and S(SL)⊆SL.
Thus,LS andSLare left ideals.
(iii) Again
S(LS) =S(SL)⊆SL and S(SL) =L(SS)⊆LS.
Hence,LS andSLare left connected. Also,
(SL)S ⊆LS and (LS)S= (SS)L⊆SL.
Thus,SLandLS are right connected, too. 2
Lemma 5 part(i)provided a guarantee for left idealLof CA-AG-groupoidS andM ⊆S that LM will surely be a left ideal ifLM = M L. However, ifLM 6= M Lthen it does not mean that LM will never be a left ideal. We provide example to verify that ifLM 6= M LthenLM may or
may not be a left ideal.
The above example clarify that ifM L⊂LM thenLM is a left ideal and if neitherM L⊂LM
norLM ⊂M L, thenLMis not a left ideal.
As by part(ii)of Lemma 5, for left idealL,LSandSLare left ideals. By using this result, asLS
andSLare left ideals so(LS)SandS(SL)are left ideals and so on. Also by part(iii)of Lemma 5,
for left idealL,LSandSLare connected sets. As for left idealL,LSandSLare left ideals (by part
(ii)) so(LS)SandS(SL)are connected sets. Coupling part(i)and(ii), as(LS)SandS(SL)are
left ideals so((LS)S)SandS(S(SL))are connected sets, and so on. Thus the following corollary is
now obvious.
Corollary 2 — LetLbe a left ideal of a CA-AG-groupoidSthen
(i) (LS)S and S(SL) are left ideals and generally ((((LS)S)S)...)S and S(...(S(S(SL))))are left ideals,
(ii) (LS)S and S(SL) are connected sets and generally ((((LS)S)S)...)S and S(...(S(S(SL))))are connected sets.
Lemma 6 — LetLbe a left ideal of a CA-AG-groupoidS andM ⊆ S such thatM L ⊂ LM,
thenLM is a left ideal ofS.
PROOF: By cyclic associativity and given condition S(LM) = M(SL) ⊆ M L ⊂ LM, thus
LM is left ideal ofS. 2
Remark 1 : IfMandNare left connected thenM2=M M⊆SM ⊆N ⇒M2 ⊆N. Similarly, N2⊆M. Again, ifMandN are right connected then M2⊆N andN2 ⊆M.
As proved in Lemma 1 that for right idealRof CA-AG-groupoid SandM ⊆ S,M Ris a left
ideal. However,M Rmay not be a right ideal, similarlyRM may not necessarily be a left or right
ideal. In the following lemma, we prove that in case of CA-AG*-groupoidM RandRM are ideals
ofS and ifR(right ideal) is replace by left idealLthenLM andM Lare connected sets.
Theorem 2 — LetR,Lare respectively right and left ideals of CA-AG*-groupoidSandM ⊆S.
Then
(i) M RandRM are ideals ofS,
(ii) M RandRM are connected sets,
PROOF:
(i) By cyclic associativity
S(M R) =R(SM) =M(RS)⊆M R⇒S(M R)⊆M R.
Hence,M Ris a left ideal ofS. Again, by left invertive law, AG* and cyclic associativity
(M R)S= (SR)M =R(SM) =M(RS)⊆M R⇒(M R)S⊆M R.
Thus,M Ris a right ideal and hence an ideal. Now, by definition of AG*
S(RM) = (RS)M ⊆RM ⇒S(RM)⊆RM.
Thus,RM is a left ideal. Again, by left invertive law, AG* and cyclic associativity
(RM)S = (SM)R=M(SR) =R(M S) =S(RM) = (RS)M ⊆RM ⇒(RM)S ⊆RM.
Thus,RM is right ideal, too. Hence,RM is an ideal ofS.
(ii) By left invertive law, AG* and cyclic associativity
S(M R) = (M S)R= (RS)M ⊆RM,
and S(RM) =M(SR) =R(M S) = (M R)S
= (SR)M =R(SM) =M(RS)⊆M R.
Thus,M RandRM are left connected. Again
(M R)S=R(M S) =S(RM) = (RS)M ⊆RM,
and (RM)S=M(RS)⊆M R.
Thus,M RandRM are right connected.
(iii) By cyclic associativity and AG*
S(M L) =L(SM) = (SL)M ⊆LM ⇒S(M L)⊆LM,
and S(LM) =M(SL)⊆M L⇒S(LM)⊆M L.
Hence,M LandLM are left connected. Also, by left invertive law and AG*
(M L)S = (SL)M ⊆LM ⇒(M L)S ⊆LM,
Thus,M LandLM are right connected. 2
Lemma 7 — LetSbe a CA-AG-groupoid andM ⊆S, then
(i) M Sis left ideal,
(ii) M SandSM are ideals ifSis AG*.
PROOF:
(i) By cyclic associativity
S(M S) =S(SM) =M(SS)⊆M S⇒S(M S)⊆M S.
Thus,M Sis left ideal.
(ii) As by part(i)M Sis left ideal. Now, we proceed to prove thatM Sis right ideal. By AG* and
cyclic associativity
(M S)S=S(M S) =S(SM) =M(SS)⊆M S.
Hence,M Sis also a right ideal and hence an ideal ofS. Now, we prove thatSM is an ideal.
By AG* and cyclic associativity
S(SM) = (SS)M ⊆SM,
and (SM)S=M(SS) =S(M S) =S(SM) = (SS)M ⊆SM.
Hence,SMis an ideal. 2
As in part (ii) of Lemma 5, we proved that for left idealL of CA-AG-groupoidS, LS is left
ideal ofS, here in Lemma 7 part(i)we proved that for anyM ⊆S,M Sis left ideal, indeed it is the
generalization of preceding mention one.
Lemma 8 — LetSbe a CA-AG*-groupoid andM ⊆S, thenM SandSM are connected.
PROOF: By cyclic associativity and AG*
S(M S) =S(SM) = (SS)M ⊆SM and S(SM) =M(SS)⊆M S.
Thus,M SandSMare left connected. Again, by left invertive law and AG*
(M S)S = (SS)M ⊆SM and (SM)S=M(SS)⊆M S.
As in CA-AG*-groupoidS, forM ⊆S,M S andSM are ideals (subsets) ofS, thus from part
(ii) of Lemma 7,(M S)S andS(SM)are ideals of S and on similar way((((M S)S)S)...)S and S(...(S(S(SM))))are ideals ofS. Similarly, by using Lemma 8,(M S)SandS(SM)are connected
sets and generally((((M S)S)S)...)SandS(...(S(S(SM))))are connected sets. Thus, the following
corollary is obvious.
Corollary 3 — LetS be a CA-AG*-groupoid andM ⊆S, then
(i) (M S)S and S(SM) are ideals of S and generally ((((M S)S)S)...)S and S(...(S(S(SM))))are ideals,
(ii) (M S)S and S(SM) are connected sets and generally ((((M S)S)S)...)S and S(...(S(S(SM))))are connected sets.
Example 1 shows that in CA-AG-groupoid, right and left ideals are distinct. However, in case of
CA-AG-band, right and left ideals coincide, as proved in the following.
Theorem 3 — LetSbe a CA-AG-band andM ⊆S. ThenMis right ideal iffM is left ideal.
PROOF: LetM is right ideal.Then by left invertive law and definition of AG-band
SM = (SS)M = (M S)S ⊆M S⊆M ⇒SM ⊆M.
Hence, M is left ideal. Conversely, suppose M is left ideal, then by cyclic associativity and
definition of AG-band
M S=M(SS) =S(M S) =S(SM)⊆SM ⊆M ⇒M S ⊆M.
Hence,M is right ideal. 2
Theorem 4 — LetLbe a left ideal of CA-T1-AG-groupoidSandN ⊆S, then
(i)LN is right ideal,
(ii)N Lis left ideal.
PROOF: (i) By cyclic associativity, left invertive law and definition ofT1-AG-groupoid
(LN)S = (SN)L⇒S(LN) =L(SN) =N(LS)
⇒(LN)S = (LS)N = (N S)L
⇒S(LN) =L(N S) =S(LN) =N(SL)
Hence,LN is a right ideal.
(ii) Again, by cyclic associativity, left invertive law and definition ofT1-AG-groupoid
S(N L) =L(SN) =N(LS)
⇒(N L)S = (LS)N = (N S)L
⇒S(N L) =L(N S) =S(LN) =N(SL)⊆N L.
Hence,N Lis left ideal. 2
Theorem 5 — LetRbe a right ideal of a CA-T1-AG-groupoidSandN ⊆S, thenRN andN R are right connected.
PROOF: By cyclic associativity and definition ofT1-AG-groupoid
(M N)S = (SN)M ⇒S(M N) =M(SN) =N(M S)⊆N M,
and (N M)S = (SM)N ⇒S(N M) =N(SM) =M(N S)
⇒(N M)S = (N S)M = (M S)N ⊆M N.
Hence,RN andN Rare right connected. 2
Lemma 9 — LetM andN are left connected sets of AG-groupoidS andM is left ideal ofS.
ThenS(M∪N)⊆M, consequentlyM∪N is a left ideal.
PROOF: Letst∈ S(M ∪N), wheres∈ S andt ∈M ∪N.Thent∈ Mort∈ N.Ift ∈M
thenst∈SM ⊆M. Also ift∈N,thenst∈SN ⊆M.Hence,S(M∪N)⊆M, asM ⊆M∪N,
henceM∪N is left ideal. 2
Lemma 10 — LetM andN are left connected sets of AG-groupoidS, thenM∪N andM∩N
are left ideals.
PROOF: Letst∈S(M∪N), wheres∈Sandt∈M∪N.Thent∈Mort∈N.Ift∈M then st∈SM ⊆N. Also ift∈N,thenst∈SN ⊆M.Hence,st∈M∪N, thusS(M∪N)⊆M∪N,
henceM∪N is left ideal.
Now, we prove thatM ∩N is left ideal. Letsr ∈ S(M ∩N), wheres ∈ S andr ∈ M ∩N.
Thenr ∈M and r∈N, which impliessr∈SM ⊆N andsr∈SN ⊆M.Hence,sr ∈M∩N,
thusS(M ∩N)⊆M ∩N, henceM∩N is left ideal. 2
Lemma 11 — LetM andNare right connected sets of AG-groupoidS, thenM∪N andM∩N
PROOF: Letts∈(M∪N)S, wheret∈M∪Nands∈S. Thent∈Mort∈N.Ift∈Mthen ts∈M S⊆N. Again, ift∈N,thents∈N S⊆M.Hence,ts∈M∪N, thus(M∪N)S⊆M∪N,
henceM∪N is right ideal.
Letrs∈ (M ∩N)S, wherer ∈ M∩N ands∈ S. Thenr ∈ M and r ∈ N, which implies rs∈M S⊆Nandrs∈N S ⊆M.Hence,rs∈M∩N, thus(M∩N)S⊆M∩N, henceM∩N
is right ideal. 2
Corollary 4 — LetM andN are connected sets of AG-groupoidS, thenM∪N andM∩N are
ideals ofS.
2.2 (Left/right) Ideals associated with a fixed element of CA-AG-groupoid
In the following, we proved that in CA-AG-groupoidS for left idealL, right idealR, idealI and u∈S,uRis left ideal, while(i)Luis a left ideal ifuL=Luand(ii)uIis an ideal ifuI=Iu. We
also provided supporting and counter examples to illustrate these results. Further, we proved that ifS
is also AG* then(i)uLanduRare ideals,(ii)LuanduLare connected sets and(iii)RuanduR
are connected sets.
Lemma 12 — LetRandLrespectively be right and left ideals of a CA-AG-groupoidSandube
a fixed element ofS. Then
(i) Luis left ideal ifuL=Lu,
(ii) uLis left ideal ifuL=Lu,
(iii) uRis left ideal.
PROOF: (i) Lets∈Sandp=lu∈Lu, wherel∈L. By cyclic associativity
∪
s∈S, p∈Lu(sp) =s∈S, l∪∈L(s·lu) =s∈S, l∪∈L(u·sl)⊆uL=Lu⇒S(Lu)⊆Lu.
Hence,Luis left ideal.
(ii) Follow by part(i).
(iii) Supposes∈Sandq =ur∈uR, wherer∈R. Then
∪
s∈S, q∈uR(sq) =s∈S, r∪∈R(s·ur) =s∈S, r∪∈R(r·su) =s∈S, r∪∈R(u·rs)⊆uR⇒S(uR)⊆uR.
The following example depicts that the conditionuL=Luis necessary foruLto be a left ideal
and ifuL6=LuthenuLandLumay not be left ideals.
Example 9 : Recall Example 1,Sis CA-AG-groupoid.
(i) L={2,3,5,7}is left ideal ofS. Foru = 0∈S, Lu={3},uL={3,5}, clearlyuL6=Lu.
Now, S(uL) = {3,7} * uL, thus uLis not a left ideal of S. However, Luis left ideal as S(Lu) ={3} ⊆Lu.
(ii) R = {1,3,4,7} is right ideal of S. For u = 0,1,3,4,5,6 & 7, uR = {3} and S(uR) = {3} ⊆ uR, thusuRis left ideal foru = 0,1,3,4,5,6 & 7. Foru = 2, uR = {3,6,7}and S(uR) = {3,7} ⊆ uR, thusuRis also left ideal foru = 2. Thus,∀u ∈ S, S(uR) ⊆ uR,
henceuRis left ideal ofS ∀u∈S. Note that inspite of the fact that for
u= 0, Ru={3,4} 6={3}=uR,
u= 2, Ru={3} 6={6,7}=uR,
u= 5, Ru={3,7} 6={3}=uR,
stilluRis left ideal, thus conditionuR=Ruis not required here.
Lemma 13 — LetRandLrespectively be right and left ideals of CA-AG*-groupoidSandu∈S
be a fixed element ofS. Then
(i) uLanduRare ideals ofS,
(ii) LuanduLare connected sets,
(iii) RuanduRare connected sets.
PROOF: (i) Letul∈uLands∈S, wherel∈L, then by AG* and cyclic associativity
∪
l∈L, s∈S∪(ul·s) =l∈L, s∪∈S(l·us) =l∈L, s∪∈S(s·lu) =l∈L, s∪∈S(u·sl)⊆uL⇒(uL)S ⊆uL.
Hence,uLis right ideal. Again, by AG*, cyclic associativity and left invertive law
∪
s∈S, l∈L(s·ul) =s∈S, l∪∈L(l·su) =s∈S, l∪∈L(u·ls) =s∈S, l∪∈L(lu·s)
= ∪
Thus,uLis left ideal, too. Hence,uLis an ideal. Again, letur∈uRands∈S, wherer ∈R. Then
by left invertive law, AG* and cyclic associativity
∪
r∈R, s∈S(ur·s) =r∈R, s∪∈S(sr·u) =r∈R, s∪∈S(r·su)
= ∪
r∈R, s∈S(u·rs) =u·RS⊆uR⇒(uR)S⊆uR.
Hence,uRis right ideal. Again, by cyclic associativity
∪
s∈S, r∈R(s·ur) =s∈S, r∪∈R(r·su) =s∈S, r∪∈R(u·rs)⊆uR⇒S(uR)⊆uR.
Thus,uRis also left ideal.
(ii) Fors∈Sandlu∈Lu, wherel∈L, by left invertive law and AG*
∪
l∈L, s∈S(lu·s) =l∈L, s∪∈S(su·l) =l∈L, s∪∈S(u·sl)⊆uL⇒(Lu)S ⊆uL. (2.1)
Again, fors∈Sandul∈uL, wherel∈L,by left invertive law
∪
l∈L, s∈S(ul·s) =l∈L, s∪∈S(sl·u)⊆Lu⇒(uL)S ⊆Lu. (2.2)
From (2.1) and (2.2), LuanduLare right connected. Now to prove thatLuanduLare left
connected, lets∈Sandlu∈Lu, forl∈L. By cyclic associativity
∪
s∈S, l∈L(s·lu) =s∈S, l∪∈L(u·sl)⊆uL⇒S(Lu)⊆uL. (2.3)
Again, forul∈uL, wherel∈L,by cyclic associativity and AG*
∪
s∈S, l∈L(s·ul) =s∈S, l∪∈L(l·su) =s∈S, l∪∈L(sl·u)⊆Lu⇒S(uL)⊆Lu. (2.4)
From (2.3) and (2.4),LuanduLare left connected. 2
(iii) Supposes∈Sandur∈uR, wherer∈R. Then by AG* and cyclic associativity
∪
r∈R, s∈S(ur·s) =r∈R, s∪∈S(r·us) =r∈R, s∪∈S(s·ru) =r∈R, s∪∈S(rs·u)⊆Ru⇒(uR)S ⊆Ru. (2.5)
Again, forru∈Ru, wherer∈R, then by definition of AG*
∪
r∈R, s∈S(ru·s) =r∈R, s∪∈S(u·rs)⊆uR⇒(Ru)S⊆uR. (2.6)
From (2.5) and (2.6), uR and Ru are right connected. To prove that uR and Ru are left
connected, lets∈Sandur∈uR, forr ∈R. Then by AG* and left invertive law
∪
Again, forru∈Ru, wherer ∈R, by AG*, left invertive law and cyclic associativity
∪
s∈S, r∈R(s·ru) =s∈S, r∪∈R(rs·u) =s∈S, r∪∈R(us·r) =s∈S, r∪∈R(s·ur)
= ∪
s∈S, r∈R(r·su) =∪(u·rs)⊆uR⇒S(Ru)⊆uR. (2.8)
From (2.7) and (2.8),uRandRuare left connected. 2
Lemma 14 — Letube a fixed element of a CA-AG-groupoidSandI be an ideal ofS such that uI=Iu, thenuIis an ideal ofS.
PROOF: By Lemma 12 part(ii),uI is a left ideal ofS. To prove thatuI is right ideal, suppose
ui∈uIands∈S, wherei∈I, then by cyclic associativity
∪
i∈I, s∈S(ui·s) =i∈I, s∪∈S(si·u)⊆Iu=uI ⇒(uI)S ⊆uI.
Thus,uIis a right ideal, too. Hence, the result follows. 2
Proposition 1 — [1]. Every CA-AG-groupoid is paramedial.
Theorem 6 — IfSis a CA-AG-groupoid andu∈S, then
(i) uSis left ideal,
(ii) u(Su)is left ideal,
(iii) u(Su)and(uS)uare right connected,
(iv) u(Su)and(uS)uare connected, ifSis right alternative.
PROOF: (i) By cyclic associativity
S(uS) = ∪
w,t∈Sw(ut) =w,t∪∈St(wu) =w,t∪∈Su(tw)⊆uS ⇒S(uS)⊆uS.
Hence,uSis left ideal.
(ii) By cyclic associativity and medial law
S(u(Su)) = ∪
w,t∈Sw(u·tu) =w,t∪∈Stu·wu =w,t∪∈Stw·uu
= ∪
w,t∈Su(tw·u)⊆u(Su)⇒S(u(Su))⊆u(Su).
(iii) By left invertive law and cyclic associativity
(u(Su))S = ∪
w,t∈S(u·wu)t=w,t∪∈S(t·wu)u=w,t∪∈S(u·tw)u⊆(uS)u
⇒(u(Su))S ⊆ (uS)u. (2.9)
Again, by left invertive, proposition 1 and cyclic associativity
((uS)u)S = ∪
w,t∈S(uw·u)t=w,t∪∈Stu·uw=w,t∪∈Sw(tu·u)
= ∪
w,t∈Sw(uu·t) =w,t∪∈St(w·uu) =w,t∪∈Suu·tw
= ∪
w,t∈Swu·tu=w,t∪∈Su(wu·t) =w,t∪∈St(u·wu)
= ∪
w,t∈St(u·uw) =w,t∪∈Suw·tu=w,t∪∈Su(uw·t)
= ∪
w,t∈Su(tw·u)⊆u(Su)
⇒((uS)u)S ⊆ u(Su). (2.10)
From (2.9) and (2.10),(uS)uandu(Su)are right connected.
(iv) By part(iii),(uS)uandu(Su)are right connected. We only show that(uS)uandu(Su)are
left connected. By cyclic associativity
S((uS)u) = ∪
w,t∈Sw(ut·u) =w,t∪∈Su(w·ut) =w,t∪∈Su(t·wu)
= ∪
w,t∈Su(u·tw) =w,t∪∈Stw·uu=w,t∪∈Su(tw·u)⊆u(Su)
⇒S((uS)u) ⊆ u(Su). (2.11)
Again, by cyclic associativity, left invertive law, proposition 1 and by right alternativity
S(u(Su)) = ∪
w,t∈Sw(u·tu) =w,t∪∈Sw(u·ut) =w,t∪∈Sw(t·uu)
= ∪
w,t∈Suu·wt=w,t∪∈Stu·wu=w,t∪∈S(wu·u)t
= ∪
w,t∈S(w·uu)t=w,t∪∈S(u·wu)t=w,t∪∈S(u·uw)t
= ∪
w,t∈S(t·uw)u=w,t∪∈S(w·tu)u=w,t∪∈S(u·wt)u⊆(uS)u
⇒S(u(Su)) ⊆ (uS)u. (2.12)
3. CONCLUSION
We demonstrated that (left/right) ideals and (left/right) connected sets exist in CA-AG-groupoids.
We precisely discussed some fundamental characteristics of (left/right) ideals and (left/right)
con-nected sets in CA-AG-groupoids and established relations of (left/right) ideals among them and with
(left/right) connected sets. Furthermore, we established relations of (left/right) ideals associated with
fixed elements among each other and with (left/right) connected sets and investigated different
condi-tions under which these results hold. We used the modern techniques of Prover-9, Mace-4 and GAP
to produce illustrative examples, counterexamples and provide several other examples to improve the
standard of this research work.
ACKNOWLEDGEMENT
This research work is financially supported by the HEC funded project NRPU-3509. Further, we are
extremely thankful to the unknown reviewers.
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