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STUDY SOME NEW RELATIONS OF MEDIAL AND PARAMEDIAL MAGMAS WITH SEMIGROUPS,
LA(RA) AND LDD(RDD) SEMIGROUPS
Syed Aleem Shah, Aleem Syed, Nisar Shah
To cite this version:
Syed Aleem Shah, Aleem Syed, Nisar Shah. STUDY SOME NEW RELATIONS OF MEDIAL AND PARAMEDIAL MAGMAS WITH SEMIGROUPS, LA(RA) AND LDD(RDD) SEMIGROUPS:
SOME NEW STUDY IN GROUPOIDS SATISFYING MEDIAL, PARAMEDIAL, LDD, RDD, LEFT INVERTIVE AND RIGHT INVERTIVE LAWS WITH THEIR RELATION. 2020. �hal-02424676v3�
NEW RESULTS IN MEDIAL AND PARAME- DIAL GROUPOIDS
Syed Aleem Shah, Nisar Ahmad
1984aleem@gmail.com and dr.nisarahmad@kust.edu.pk
Institute of Numerical Sciences, Kohat University of Science and Technology (KUST), KPK, Pakistan
Corresponding Author Email: 1984aleem@gmail.com
Abstract: In this article we developed the concept that on what conditions a groupoid that satisfies medial law i.e. ∀ a, b, c and d ∈ groupoid S the con- dition (ab)(cd) = (ac)(bd) becomes right double displacement semigroup (RDD- semigroup), left double displacement semigroup (LDD-semigroup), commutative groupoid and commutative semigroup. We explained the connection of RDD- semigroup and LDD-semigroup with semigroup. We developed concept of further conditions on left almost semigroup (LA-Semigroup) and right almost semigroup (RA-Semigroup) that becomes commutative semigroup. We extended our work on RDD-semigroups and LDD-semigroups in [14]; and discussed the relation of paramedial groupoid with left almost semigroup (LA-Semigroup) and right al- most semigroup (RA-semigroups), semigroup and commutative groupoid with the- orems.
Key Words: Medial; Paramedial; RDD-semigroup; LDD-semigroup; LA-Semigroup;
RA-Semigroup; LA-Monoid; RA-Monoid.
Preliminaries: In literature a groupoid S is called left almost semigroup (LA- Semigroup) [3], [6], [7]; is an algebraic structure that holds left invertive law i.e.
if ∀ a, b and c ∈ S the condition (ab)c = (cb)a holds. LA-Semigroup is also called Abel Grassmann’s Groupoid abbreviated as AG-Groupoid in [9] and in [8]
the same structure is called left invertive groupoid. A groupoid S is called right almost semigroup (RA-Semigroup) [3]; that holds right invertive law i.e. if ∀ a, b and c ∈ S the condition a(bc) = c(ba) is satisfied. A groupoid S is called medial [2], [12]; if ∀ a, b, c and d ∈ S the condition (ab)(cd) = (ac)(bd), this property (ab)(cd) = (ac)(bd) is called medial law or bisymmetry law. A semigroup S is called E-semigroup [1], [4]; if all the idempotents of S also form semigroup. A semigroup S is called regular semigroup [1], [4], [5]; if for each a ∈ S ∃ b ∈ S such that aba =a and bab =b. A semigroup S is called Orthodox semigroup [1], [4], [5]; if S is E-semigroup as well as regular semigroup. A groupoid S is called locally associative if∀ a ∈ S condition a2a=aa2 is satisfied. If each element a ∈ groupoid S is idempotent then S is locally associative groupoid. LA-Semigroup S is called LA-Monoid if S contains left identity si such that∀ sj ∈S the condition sisj = sj is satisfied and S is LA-Group if S contains left identity si such that
∀ sj ∈ S the condition sisj = sj is satisfied and inverse of each element exists.
RA-Semigroup S is called RA-Monoid if S contains right identity si such that ∀ sj ∈ S the condition sjsi = sj is satisfied and S is LA-Group if S contains right
identity si such that ∀ sj ∈ S the condition sjsi = sj is satisfied and inverse of each element exists. A groupoid S is called paramedial [12] if ∀ a, b, c and d
∈ S the condition (ab)(cd) = (db)(ca) holds. A groupoid S is called left double displacement semigroup (LDD-semigroup) [10]; if∀a, b, c and d∈S the condition (ab)(cd) = (cb)(ad) is satisfied and by the same way a groupoid S is called right double displacement semigroup (RDD-semigroup) [10]; if ∀ a, b, c and d ∈ S the condition (ab)(cd) = (ad)(cb) is satisfied.
Introduction: J.L. Chrislock [2]; elaborated that on what conditions a semigroup S becomes medial. Clifford and Preston [1]; and J.M. Howie [4]; wrote compre- hensive books on semigroup theory and discussed the details of E-Semigroups, left regular, right regular, regular, inverse semigroups, left(right) ideals and both (left and right) sided ideals of most kinds of semigroups with general theorems in com- plete detail. Kazim and Naseeruddin [3]; introduced the concept of LA-Semigroups in 1972. Mushtaq and Yousaf [6]; extended the results of Kazim and naseeruddin and discussed locally associative LA-Semigroups. Q. Mushtaq [7]; discussed that on what conditions LA-Semigroups becomes commutative group. Mushtaq and Kamran [9]; proved that “A cancellative AG-Groupoid (LA-Semigroup) G is a commutative semigroup if f orall a, b and c ∈ G the condition a(bc) = (cb)a is satisfied. P. Holgate [8]; used the different conditions to check that on what condi- tions a groupoid S satisfies the the associative law. Ahmad et al. [10]; developed new concept of LDD-semigroup and RDD-semigroup. Cho et al. [12] explained and proved that if a groupoid S is paramedial then S is medial if (a) S is unipotent and left(right) cancellative, (b) S contains left(right) neutral element (c) S is com- mutative, (d) S is left(right) modular or (e) S is idempotent. Madad et al. [13];
proved that if S is LA-monoid then S is paramedial and also holds the property that ∀ a, b, c and d∈ S the condition (ab)(cd) = (dc)(ba).
We used these concepts and definitions explained in [1] to [8] and [13] and con- structed new theorems and examples on RDD-semigroups, LDD-semigroups in [14]; also developed the concept that on what conditions a medial becomes RDD- semigroup, LDD-semigroup and semigroup becomes medial and RDD-semigroup, medial and LDD-semigroups. In [14]; we proved that if a groupoid S is paramedial, medial and either RDD-semigroup or LDD-semigroup then S is commutative. We also proved in [14]; that if S is LA-monoid and LDD-semigroup or RA-Monoid and RDD-semigroup then S is commutative monoid.
In this article we extended our work explained in [14]; and divided this article in four sections. In section-1 we discussed the further relations and conditions of medial with right double displacement semigroup (RDD-semigroup) and left double displacement semigroup (LDD-semigroup). In section-2 we explained con- ditions when a groupoid becomes commutative groupoid and commutative semi- group. In section-3 we discussed when groupoid becomes commutative medial, right(left) monoid and commutative monoid, connection of RDD-semigroup with right monoid and LDD-semigroup with left monoid and in Section-4 we developed new theorems on what conditions Paramedial groupoids and medial groupoids with left and right identities becomes commutative monoid respectively but first we give some short details of solved open problem in [14] and already proved fol- lowing results in [3] to [13] so readers can grasp the approach of this research in effortless way.
Open Problem in [10]: Here we only study about LDD-semigroup where RDD- semigroup is left as open problem for Researchers.
Answer To Open Problem in [10] explained in [14]:
Every LOn or LZ is RDD-semigroup as well as semigroup because when we have non empty set S and binary operation is defined on S by such way that ∀ a and b ∈ S, ab = a then by the definition (ab)c = ac = a and also a(bc) = ab = a.
This also holds medial law because when ab = a so we have (ab)(cd) = ac = a, (ac)(bd) =ab=a and also (ad)(cb) =ac=a.
Every ROn and RZ is semigroup, medial and LDD-semigroup because when we have non empty set S and binary operation is defined on S by ab = b ∀ a and b ∈ S then by the definition (ab)c = bc = c and also a(bc) = ac = c. This is also medial because (ab)(cd) = bd = d and (ac)(bd) = bd = d also satisfies the LDD-law because (cb)(ad) = bd=d.
Recall Examples In [14]: On R+ if we define binary operation in such way that ∀ a and b ∈ R+ ab = ln(a), ab = ea, ab = a2 and ab = ak where k 6= 1 then w.r.t these binary operations R+ is medial and RDD-semigroup but R+ is not semigroup, neither LA-Semigroup nor RA-semigroup.
Recall Examples In [10]: On R+ if we define binary operation in such way that ∀ a and b ∈ R+ ab = ln(b), ab = eb, ab = b2 and ab = bk where k 6= 1 then w.r.t these binary operations R+ is medial and LDD-semigroup but R+ is not semigroup, neither LA-Semigroup nor RA-semigroup.
Next we have already proved results which are following:
PROVED RESULTS-1 IN [3] to [13]:
If a groupoid S satisfies:
(a) left invertive law and associative law then S is commutaive and also satisfies right invertive law.
(b) right invertive law and associative law then S is commutaive and also satisfies left invertive law.
(c) commutative and left invertive law then S is associative and also satisfies right invertive law.
(d) commutative and right invertive law then S is associative and also satisfies left invertive law.
(e) commutative and associative law then S holds left invertive as well as right invertive law.
(f) paramedial law then S is medial if S atleast one of these conditions (i) S is com- mutative, (ii) S is unipotent and left(right) cancellative, (iii) S is LA-semigroup with left identity, (iv) S is left(right) modular groupoid, (v) S contains left(right) neutral element or (vi) S is idempotent.
PROVED RESULTS-2 IN [3] to [13] AND IN [14]:
If a groupoid S is:
(a) LA-semigroup with right identity then S is commutative monoid.
(b) RA-semigroup with left idedntity then S is commutative monoid.
(c) LDD-semigroup with right identity then S is commutative monoid.
(d) RDD-semigroup with left identity then S is commutative monoid.
(e) Medial with identity then S is commuttaive monoid.
(f) Paramedial with identity then S is commuttaive monoid.
(g) Paramedial, medial and LDD-semigroup then S is commutative.
(h) Paramedial, medial and RDD-semigroup then S is commutative.
(i) LA-monoid and LDD-semigroup then S is commutative monoid.
(j) RA-monoid and RDD-semigroup then S is commutative monoid.
Note: RESULTS-2 (g) to (j) are proved in [14].
Section-1 Relation of Medial With RDD-semigroup and LDD-semigroup:
In this section we discuss new theorems that on what conditions medial becomes RDD-semigroup and LDD-semigroup by the following theorems:
Theorem 1: If a groupoid S that holds medial law and satisfes condition that ∀ a, b, c and d∈ S, (ab)(cd) = (ac)(db) then S is RDD-semigroup.
Proof: By using the given conditions we have (ab)(cd) = (ac)(db) = (ad)(cb).
Theorem 2: If a groupoid S holds medial law and satisfies the condition that ∀ a, b, c and d∈ S, (ab)(cd) = (ad)(bc) then S is RDD-semigroup.
Proof: By using the given conditions we do the following steps:
(ab)(cd) = (ad)(bc) = (ab)(dc) by using condition and medial law this is proved that (ab)(cd) = (ab)(dc).
So (ab)(cd) = (ab)(dc) = (ad)(bc) = (ad)(cb) which is the required result.
Theorem 3: If a groupoid S holds medial law and satisfies the condition that∀a, b, c and d∈S, (ab)(cd) = (ba)(cd) and (ab)(cd) = (dc)(ab) then S is commutative.
Proof: Using the conditions we do the following steps:
(ab)(cd) = (ba)(cd) = (cd)(ab). Hence this is proved generally that ∀ a, b, c and d ∈S the condition (ab)(cd) = (cd)(ab) is always satisfied.
Corollary-1 Corollary Related To Theorems 1 to 3:
(a) If S is medial and RDD-semigroup then ∀ a, b, c and d ∈ S, (ab)(cd) = (ab)(dc) = (ad)(bc) = (ac)(db).
(b) If S is RDD-semigroup and satisfies the property∀a, b, c and d∈S, (ab)(cd) = (ab)(dc) then S is medial and also satisfies the properies (ab)(cd) = (ad)(bc), (ab)(cd) = (ac)(db).
Proof(a): Straightforward.
(b): Straightforward.
Theorem 4: If a groupoid S satisfies these two conditions that (ab)(cd) = (ac)(db) and (ab)(cd) = (ab)(dc),∀a, b, c and d∈S then S holds medial law and RDD-law.
Proof: By using the given conditions we have (ab)(cd) = (ac)(db) = (ac)(bd) by using (ab)(cd) = (ab)(dc) so S is medial.
Now using these conditions we have (ab)(cd) = (ab)(dc) = (ad)(bc) = (ad)(cb) by
using medial law and the condition (ab)(cd) = (ab)(dc).
Theorem 5: If S is medial and satisfies the condition (ab)(cd) = (ca)(bd) then S is LDD-semigroup.
Proof: By using the given conditions we gave (ab)(cd) = (ca)(bd) = (cb)(ad).
Hence S is LDD-semigroup.
Theorem 6: If S is medial and satisfies the property (ab)(cd) = (bc)(ad) then S is LDD-semigroup.
Proof: From the conditions we have (ab)(cd) = (ac)(bd) = (cb)(ad).
Theorem 7: If ∀ a, b, c and d ∈ groupoid S the conditions (ab)(cd) = (bc)(ad) and (ab)(cd) = (ba)(cd) are satisfied then S is medial, LDD-semigroup and also holds the property (ab)(cd) = (ca)(bd).
Proof: From the given conditions (ab) = (bc)(ad) = (cb)(ad), so S is LDD- semigroup. So we have following steps:
(ab)(cd) = (bc)(ad) = (ac)(bd) = (ca)(bd), by applying the LDD-law and the condtion (ab)(cd) = (ba)(cd) we proved the theorem.
Remarks-1 (Remarks Related To Theorems 1 to 7):
(a) If ∀ a, b, c and d ∈ S, (ab)(cd) = (ac)(db) then (ab)(cd) = (ad)(bc) and vice versa.
(b) If ∀ a, b, c and d ∈ S, (ab)(cd) = (ca)(bd) then (ab)(cd) = (bc)(ad) and vice versa.
(c) If ∀ a, b, c and d ∈ S, (ab)(cd) = (ab)(dc) = (ba)(cd) = (ba)(dc) = (cd)(ab) = (dc)(ab) = (dc)(ba) = (cd)(ba) then S is commutaive but this does not means that S satisfies associative, medial, rdd or ldd properties elaborated by Examples-1(a) and 1(b).
(d) If a groupoid S satisfies medial law and commutative law then S is surely RDD-semigroup well as LDD-semigroup and paramedial groupoid proved in [12]
and in [14] and this is trivial.
Example-1(a): Finite Commutative Magmas(Groupoids) Which Are Not Medial, Neither RDD-semigrpoup Nor LDD-Semigroup
Table 1: Table For Example 1(a)
∗ a b . a b
a b a a b b
b a a b b a
Example-1(b): Infinite Commutative groupoid Which Is Not Medial, Neither RDD-semigroup, LDD-Semigroup
On R(set of real numbers),Q(set of rational numbers) and on Z (set of integers) if ab = a.b+k where k 6= 0 then R, Q and Z w.r.t this binary operation are commuttaive groupoids but do not satisfy medial law, rdd(ldd)-law and also does not satisfy associative law.
Open Problem 1: We left this to researchers as an open problem to construct a non commutative and non associative groupoid S that satisfies only two conditions (ab)(cd) = (ac)(db) = (ad)(bc) and does not satisfy medial law, paramedial law and RDD-law.
Open Problem 2: We left this to researchers as an open problem to construct a non commutative and non associative groupoid S that satisfies only two conditions (ab)(cd) = (ca)(bd) = (bc)(ad) and does not satisfy medial law, paramedial law and LDD-law.
Section-2 Conditions When groupoid Becomes Commutative groupoid:
In this section we give connection and relation of groupoid with commutative groupoid, LDD-semigroup and RDD-semigroup with the follwoing theorems:
Theorem 8: If ∀ a, b, c and d∈ groupoid S, (ab)(cd) = (ac)(db) and (ab)(cd) = (ca)(bd) then S is commutative and holds RDD-law, LDD-law, medial law and paramedial law.
Proof: In Remarks-1(a) this is given that if ∀ a, b, c and d ∈ groupoid S, (ab)(cd) = (ac)(db) then (ab)(cd) = (ad)(bc) and if (ab)(cd) = (ca)(bd) then (ab)(cd) = (bc)(ad), so we have (ab)(cd) = (ac)(db) = (cd)(ab).
Also we can say that (ab)(cd) = (ac)(db) = (da)(cb) = (cd)(ab) that shows that S is commutative so surely (ab)(cd) = (ca)(bd) = (ac)(bd) that shows that S is medial and commutative medial is RDD-semigroup as well as LDD-semigroup.
Theorem 9: If groupoid S satisfies rdd-law and also satisfies the condition (ab)(cd) = (bc)(ad), ∀ a, b, c and d∈S then S is commutative.
Proof: Method-1: Using the given conditions we have (ab)(cd) = (ad)(cb) = (dc)(ab) = (db)(ac) = (ba)(cd) = (bd)(ca).
So (ab)(cd) = (dc)(ab) = (ba)(cd) so (ab)(cd) = (dc)(ab) = (cd)(ab).
Theorem 10: if a groupoid S satisfies ldd-law and also satisfies the condition (ab)(cd) = (ac)(db), ∀ a, b, c and d∈S then S is commutative.
Proof: Using the given coinditions we have (ab)(cd) = (cb)(ad) = (ca)(db) = (da)(cb) = (dc)(ba), so we can use these properties and prove our result or there is also second method which is given below:
If ∀ a, b, c and d ∈ S the condition (ab)(cd) = (ac)(db) is satisfied then surely (ab)(cd) = (ad)(bc) and we do the following steps to prove our result.
(ab)(cd) = (ad)(bc) = (bd)(ac) = (bc)(da) = (ba)(cd) = (ca)(bd) = (cd)(ab).
Theorem 11: If groupoid S satisfies one condtion from (ab)(cd) = (ad)(bc) = (ac)(db) = (ad)(cb) and one condition from (ab)(cd) = (bc)(ad) = (ca)(bd) = (cb)(ad) then S is commutative medial and satisfies all of these conditions.
Proof: Straightforward by using theorems −8 to 10.
Remarks-2 (Remarks Related To Theorems 8 to 11):
(a) If ∀a, b, c and d ∈groupoid S, (ab)(cd) = (db)(ac) then S holds the property (ab)(cd) = (cb)(da) and vice versa.
(b) If∀ a, b, c and d∈ groupoid S, (ab)(cd) = (bd)(ca) then S holds the property (ab)(cd) = (da)(cb) and vice versa.
Lemma 1: If ∀ a, b, c and d ∈ groupoid S, (ab)(cd) = (ca)(db) then S satis- fies these conditions also (ab)(cd) = (dc)(ba) and (ab)(cd) = (bd)(ac) and similarly if If ∀ a, b, c and d ∈groupoid S, (ab)(cd) = (bd)(ac) then S satisfies these condi- tions such that (ab)(cd) = (dc)(ba) and (ab)(cd) = (ca)(db).
Theorem 12: If ∀ a, b, c and d ∈ groupoid S, medial law and condition (ab)(cd) = (ca)(db) or (ab)(cd) = (bd)(ca) then S is commutative.
Proof: Straightforward by using the conditions (ab)(cd) = (ca)(db) = (cd)(ab) and (ab)(cd) = (ac)(bd) = (cd)(ab).
Note In Theorem 12: If ∀ a, b, c and d ∈ groupoid S medial law is satis- fied and condition (ab)(cd) = (dc)(ba) is satisfied then S is paramedial but this does not mean that S is commutative e.g. RA-Monoid and LA-Monoid is medial as well as paramedial and also satisfies condition (ab)(cd) = (dc)(ba).
Lemma 2: If∀a, b, c and d∈ groupoid S the conditions (ab)(cd) = (ab)(dc) and (ab)(cd) = (ba)(cd) hold then (ab)(cd) = (ba)(cd) = (ba)(dc).
Theorem 13: If ∀ a, b, c and d ∈ groupoid S the condition (ab)(cd) = (dc)(ba) is satisfied and any condition explained in lemma-2 is satisfied then S is commu- tative.
Proof: Using the given conditions we have (ab)(cd) = (dc)(ba) = (cd)(ba) = (cd)(ab) that shows that S is commutative groupoid.
Lemma 3: If∀a, b, c and d∈groupoid S, (ab)(cd) = (da)(bc) then S is commut- taive because when we again apply the given condition we get (da)(bc) = (cd)(ab).
Also S satisfies medial, paramedial, rdd-law and ldd-law.
Proof: Straightforward.
Lemma 4: If∀a, b, c and d∈groupoid S, (ab)(cd) = (bc)(da) then S is commut- taive because when we again apply the given condition we get (bc)(da) = (cd)(ab).
Also S satisfies medial, paramedial, ldd-law and rdd-law if S satisfies the condition (ab)(cd) = (bc)(da).
Proof: Straightforward.
Remarks-3: Remarks Related To Lemmas 3 and 4:
The properies elaborated in lemmas 3 and 4 are stronger commutative properties because if groupoid S satisfies properties elaborated in lemma-3 and in lemma-4 then S is commutative and S also holds medial, right double displacement (rdd), left double displacement (ldd) and paramdial properties. We explain Remarks 2 and 3 and Lemmas 1 to 4 by the following examples:
Example-2(a): On sets of real numbers and on rational numbers if we define binary operation in such way that ab= (a+b)/2, ab= (a+b)/3,ab= (a+b)/k where k 6= 0 and 1 then R and Q are commutative magmas and satisfies medial, paramedial, left(right) double displacement properties and satisfies all the con- ditions discussed in Remarks-2 and in Lemma-1 to 3 but R and Q do not hold associative, left inveruve and right invertive laws.
Note: Example-1(a) and Example-1(b) are examples when groupoid is commu- tative but does not satisfies any property in Remarks-2 and lemmas 1 to 4.
Theorem 14: If S is RDD-semigroup and holds condition (ab)(cd) = (dc)(ba) then S is commutative.
Proof: Using the given conditions (ab)(cd) = (ad)(cb) = (bc)(da) and by lemma-4 S is commutative.
Alternate Proof: (ab)(cd) = (ad)(cb) = (bc)(da) = (ba)(dc) = (cd)(ab) which shows that S is commuttaive.
Corollary Related To Theorem 13:
If S is RDD-semigroup, semigroup and holds condition (ab)(cd) = (bc)(ad) then S is commutative semigroup as well as LA-Semigroup and RA-Semigroup.
Proof: Straightforward by using Theorem-13 and Results-1(e).
Theorem 15: If S is LDD-semigroup and holds condition (ab)(cd) = (dc)(ba), ∀ a, b, c and d∈ S then S is commutative.
Proof: Using the given conditions we do the following steps:
(ab)(cd) = (cb)(ad) = (da)(bc) and by lemma-3 S is commutative.
Alternate Proof: (ab)(cd) = (cb)(ad) = (da)(bc) = (ba)(dc) = (cd)(ab) that shows that S is commutative.
Corollary Related To Theorem 14:
(a) If S is RDD-semigroup, semigroup and holds the condition (ab)(cd) = (dc)(ba),
∀a, b, c and d ∈ S then S commuttaive semigroup and surely LA-Semigroup and RA-Semigroup.
(b) If S is LDD-semigroup, semigroup and holds the condition (ab)(cd) = (dc)(ba),
∀a, b, c and d ∈ S then S commuttaive semigroup and surely LA-Semigroup and RA-Semigroup.
Proof: Using the Theorems 14 and 15 and Results-1(e) this is straightforward.
Theorem 16: If S is medial and holds condition (ab)(cd) = (bd)(ca), ∀ a, b, c and d∈ S then S is commutative.
Proof: Using the conditions ∀ a, b, c and d ∈ S we have (ab)(cd) = (ac)(bd) = (cd)(ba). So we have the following steps:
(ab)(cd) = (cd)(ba) = (cb)(da) = (ba)(dc) = (bd)(ac), by using medial law and given condition.
So (ab)(cd) = (ac)(bd) = (cd)(ab) that shows that S is commutative.
Theorem 17: If a groupoid S holds the conditions (ab)(cd) = (bd)(ac) and
(ab)(cd) = (ab)(dc) for all a, b, c and d ∈ S then S is commutative and S is medial, LDD-semigroup and RDD-Semigroup.
Proof: Using the given conditions we have two methods by the following steps:
(ab)(cd) = (bd)(ac) = (bd)(ca) = (da)(bc), so we can use lemma-3 and this is quite obvious that S is commutative.
Alternate Proof: (ab)(cd) = (bd)(ac) = (bd)(ca) = (da)(bc) = (ac)(db) = (ac)(bd) that shows that S is medial.
So (ab)(cd) = (ac)(bd) = (ac)(db) = (ad)(cb) that shows that S is RDD-semigroup.
So (ab)(cd) = (ad)(cb) = (db)(ac) = (dc)(ab) = (cb)(da) = (cb)(ad) that shows that S is LDD-semigroup.
So if S is RDD-semigroup, LDD-semigroup then S is surely commutative.
Theorem 18: If a groupoid S holds the conditions (ab)(cd) = (bd)(ac) and (ab)(cd) = (ba)(cd), ∀ a, b, c and d ∈ S then S is commutative and S is me- dial, RDD-semigroup and LDD-Semigroup.
Proof: Using the given conditions we have the following steps:
(ab)(cd) = (bd)(ac) = (dc)(ba) = (cd)(ba) by using conditons (ab)(cd) = (bd)(ac) and (ab)(cd) = (ba)(cd).
So (ab)(cd) = (cd)(ba) = (da)(cb) = (ad)(cb) which shows that S is RDD- semigroup and also (ab)(cd) = (dc)(ba) = (ca)(db) = (ac)(db) = (cb)(ad) which shows that S is LDD-semigroup.
So if S is RDD-semigroup as well as LDD-semigroup then surely S is comuta- tive and S also satisfies medial and paramedial law.
Theorem 19: If a groupoid S holds conditions (ab)(cd) = (ca)(db) and (ab)(cd) = (ba)(cd),∀ a, b, c and d ∈S then S is commutative.
Proof: Using the given conditions we have (ab)(cd) = (ca)(db) = (ac)(db) and also (ab)(cd) = (ca)(db) = (dc)(ba) therefore∀a, b, c and d lies in S the conditions (ab)(cd) = (ca)(db) = (dc)(ba) = (cd)(ba) = (ac)(db) are satisfied.
So (ab)(cd) = (ca)(db) = (bd)(ac) = (db)(ac) = (ad)(cb) = (da)(cb) which shows that S is RDD-semigroup and also satisfies the condition (ab)(cd) = (da)(cb).
So (ab)(cd) = (da)(cb) = (db)(ca) = (cd)(ab) by applying right double displace- ment law (RDD-law) and the condition (ab)(cd) = (ca)(db).
Theorem 20: If a groupoid S is medial and holds the condition (ab)(cd) = (cb)(da) then S is commutative.
Proof: ∀a, b, c and d∈S (ab)(cd) = (ac)(bd) and (ab)(cd) = (cb)(da) so we have (ab)(cd) = (ac)(bd) = (bc)(da) and by lemma-3 S is commutative.
Alternate Proof: (ab)(cd) = (cb)(da) = (cd)(ba) = (bd)(ac). Therefore (ab)(cd) = (ac)(bd) = (cd)(ab) that shows that S is commutative.
Section-3 Relation of Groupoid, RDD-semigroup and LDD-semigroup With Right Monoid, Left Monoid and Commutative Monoid:
This is already proved in [10] and in [14] that LDD-semigroup with right iden- tity is commutative monoid and RDD-semigroup with left identity is commutative monoid. Now we show that on what conditions a Groupoid, RDD-semigroup and LDD-semigroup becomes right monoid, left monoid and commutative monoid by the following theorems:
Theorem 21: If S is RDD-semigroup with right identity and holds condition (ab)c= (ac)b, ∀ a, b and c ∈ S then S is right monoid.
Proof: ∀ a, b and c ∈ S we have (ab)c= (ac)b= (ac)(be) = (ae)(bc) =a(bc). So S holds associative law and S is right monoid.
We can also use the method a(bc) = (ae)(bc) = (ac)(be) = (ac)b = (ab)c because this is given that (ab)c= (ac)b.
Corollary Related To Theorem 20:
(a) If S is RDD-semigroup with right idenity then (ab)c = (ab)(ce) = (ae)(cb) = a(cb) and also we can saya(bc) = (ae)(bc) = (be)(ac) = b(ac).
(b) If S is RDD-semigroup with the conditioin (ab)c = (ac)b then (ab)(cd) = (a(cd))b.
Theorem 22: If S is LDD-semigroup with left identity then S is left monoid if ∀ a, b and c ∈ S the condition a(bc) =b(ac) is satisfied.
Proof: ∀ a, b and c ∈ S, (ab)(cd) = (cb)(ad) and also S contains left identity i.e.
∀ a∈ S ∃ e such that ea=a so we have the following steps:
(ab)c= (ab)(ec) = (eb)(ac) =b(ac) = a(bc) becausea(bc) = b(ac).
Corollary Related To Theorem 21:
(a) If S is LDD-semigroup with left idenity then (ab)c = (ab)(ec) = (eb)(ac) = b(ac) and a(bc) = (ea)(bc) = (ba)(ec) = (ba)c.
(b) If S is LDD-semigroup with the conditioin a(bc) = b(ac) then (ab)(cd) = c((ab)d).
Theorem 23: If ∀ a, b, c and d ∈ groupoid S the condition (ab)(cd) = (ad)(bc) and (ab)c=b(ca) are satisfied and S contains left identity then S is commutative.
Proof: In Remarks-1(a) if ∀ a, b, c and d ∈ S the condition (ab)(cd) = (ad)(bc) then (ab)(cd) = (ac)(db) and also this is given that (ab)c=b(ca) so by using these conditions we do the following steps:
Soab= (ee)(ab) = (eb)(ea) =ba that shows that S is commutaive so left identity is also right identity.
So (ab)c= c(ab) and (ab)c=b(ca) = (ca)b so (ab)c = (ca)b = (bc)a = (cb)a that shows that S is LA-Semigroup and also we proved that S is commutative so from Proved Results1 surely S is commutative monoid.
Theorem 24: If ∀ a, b, c and d ∈ groupoid S the condition (ab)(cd) = (ad)(bc) is satisfied and S contains right identity then S is right monoid.
Proof: Using the given conditions (ab)c = (ab)(ce) = (ae)(bc) = a(bc). Thus S
satisfies associative property and S is right monoid.
Note To Theorem 23We already discussed in Remarks-1 that if If∀a, b, c and d
∈groupoid S the condition (ab)(cd) = (ad)(bc) is satisfied then (ab)(cd) = (ac)(db) also. So the above theorem can be proved for a groupoid satisfying (ab)(cd) = (ac)(db) property.
Theorem 25: If ∀ a, b, c and d ∈ groupoid S the condition (ab)(cd) = (bc)(ad) is satisfied and S contains left identity then S is left monoid.
Proof: using the conditions we have a(bc) = (ea)(bc) = (ab)(ec) = (ab)c. Thus S is associative groupoid and hence left monoid.
Note Related To Theorem 24 We already proved that if ∀ a, b, c and d
∈ grouopoid S if (ab)(cd) = (bc)(ad) then (ab)(cd) = (ca)(bd) so above theorem can be proved for a groupoid satisfying property (ab)(cd) = (ca)(bd) and having left identity.
Theorem 26: If ∀ a, b, c and d ∈ groupoid S the condition (ab)(cd) = (bc)(ad) is satisfied and S contains right identity then S is commutative monoid.
Proof: In Remarks-1 we have if ∀ a, b, c and d ∈ groupoid S the condition (ab)(cd) = (bc)(ad) is satisfied then condition (ab)(cd) = (ca)(bd) is also satisfied.
Also (ab)c= (ab)(ce) = (bc)(ae) = (bc)a = (ca)b. Let e be right identity in S so we do the following steps:
ab = (ab)(ee) = (be)(ae) = ba because e is right identity and we have proved that S is commutative.
So (ab)c= (ba)cand also (ab)c= (bc)a= (cb)athat shows that S is LA-Semigroup with right identity and this is proved in [3] and in [6] that LA-Semigroup with right identity is commutative monoid.
Remarks-4 Remarks Related To Theorems 20 to 26:
(a) If S is RDD-semigroup or if S satisfies (ab)(cd) = (ad)(bc) = (ac)(db), ∀ a, b, c and d∈ S with left identity then S is commutative monoid.
(b) If S is LDD-semigroup or if S satisfies (ab)(cd) = (bc)(ad) = (ca)(bd), ∀ a, b, c and d∈ S with right identity then S is commutative monoid.
(c) If S is medial and satisfies the consition (ab)(cd) = (ab)(dc),∀ a, b, c and d ∈ S with right left identity then S is commutative monoid.
(d) If S is medial and satisfies the property (ab)(cd) = (ba)(cd),∀ a, b, c and d ∈ S with right identity then S is commutative monoid.
Theorem 27: If S is LA-Semigroup and holds condition (ab)(cd) = (bd)(ca),
∀ a, b, c and d∈ S then S is commutative semigroup.
Proof: We know that every LA-Semigroup satisfies medial law so we have (ab)(cd) = (bd)(ca) = (bc)(da). So we can use lemma-4 and this is quite easy that S is com- mutative semigroup.
Alternate Proof: We can also use properties by the following steps:
(ab)(cd) = (ac)(bd) = (cd)(ba) = (cb)(da) = (ba)(cd). So (ab)(cd) = (bd)(ca) =
(db)(ca) which shows that S is paramedial. So we do the following steps:
(ab)(cd) = (db)(ca) = (ba)(dc) = (bd)(ac) = (cd)(ab) by applying medial, parame- dial and (ab)(cd) = (bd)(ac) properties.
Corollary Related To Theorem 27: If S is RA-Semigroup and holds con- dition (ab)(cd) = (bd)(ca),∀ a, b, c and d ∈S then S is commutative semigroup.
Proof: Straightforward.
Theorem 28: If S is LA-Semigroup and satisfies condition (ab)(cd) = (da)(bc) then S is commutative semigroup.
Proof: Straightforward by using lemma-3 and Results-1(a).
Alternate Proof: Without using lemma-3 we will do the following steps:
(ab)(cd) = (da)(bc) = ((bc)a)d= ((ac)b)d = (db)(ac) = (cd)(ba) = (ac)(db).
S is surely medial beacuse S is LA-Semigroup so using Theorems 1 to 6 this is clear that S is RDD-semigroup as well as LDD-semigroup so S is commutative.
Also by using Theorems 8 to 11 this is straightforward that S is commutative.
Corollary Related To Theorem 28: If S is RA-Semigroup and satisfies condi- tion (ab)(cd) = (da)(bc), ∀ a, b, c and d∈ S then S is commutative semigroup.
Proof: Straightforward by using lemma-3 and Results-1.
Alternate Proof: Without using lemma-3 we do the following steps:
(ab)(cd) = (da)(bc) = c(b(da)) =c(a(db)) = (db)(ac) = (cd)(ba) = (ac)(db)
S is medial because S is RA-Semigroup so by using Theorems 1 to 6 this is clear that S is RDD-semigroup as well as LDD-semigroup so S is commutative.
Theorem 29: If S is LA-Semigroup and satisfies condition (ab)(cd) = (bc)(da),∀ a, b, c and d∈ S then S is commutative semigroup.
Proof: Straightforward by using lemma-4 and Results-1.
Alternate Proof: Without using lemma-4, we can prove this result by using the medial law and the given condition because S is LA-Semigroup then obviusly S satisfies bisymmetry law or medial law and we do the following steps:
(ab)(cd) = (bc)(da) = (bd)(ca) = (dc)(ab) = (ca)(bd) = (cb)(ad) which shows that S is LDD-semigroup.
(ab)(cd) = (cb)(ad) = (ba)(dc) = (bd)(ac) by applying ldd-law, given condition and medial law. So (ab)(cd) = (ac)(bd) = (cd)(ab)
Theorem 30: If S is RA-Semigroup and satisfies the condition (ab)(cd) = (bc)(da),
∀ a, b, c and d∈ S then S is commutative semigroup.
Proof: Straightforward by using lemma-4 and Results-1.
Alternate Proof: Without using the lemma4, we can prove required result by using medial law and given condition that we used in the proof of theorem-29.
Open Problem 3: This is left to researchers as an open problem that is this possible to construct a non commutative and non associative groupoid S that holds only medial or bisymmetry law but does not hold RDD-law, LDD-law, left invertive law, right invertive law, paramedial law and any other law or property discussed in Remarks-1 and Remarks-2.
Example-3(a): Non Assocaitive Groupoid Which Contains Identity With Inverse of Each Element But Does Not Satisfy RDD-law, LDD-law and Medial-Law
Let we have set Z5 ={0,1,2,3,4} and binary operationη defined on Z5 that we have the following table:
Table 2: Table For Example 3(a)
η 0 1 2 3 4
0 0 1 2 3 4
1 1 1 1 0 1
2 2 2 2 2 0
3 3 0 3 3 3
4 4 4 0 4 4
This is non commutative, non associative but locally associative groupoid because each element is idempotent and Z5 w.r.t binary operation η contains identity ele- ment 0 and inverse of each element exists. Clearly from table 1η2 = 1 and 2η1 = 2 and (2η2)η4 = 2ηη4 = 1 and 2η(2η4) = 2η1 = 2. Also Z5 does not satisfy left invertive law and right invertive law. This is not LDD-semigroup, RDD-semigroup and this is not medial if we see (2η4)η(3η3) = 1η3 = 3 but (2η3)η(4η3) = 2η4 = 1.
Section-4 Relation of Paramedial groupoid With Semigroup, LA-Semigroup, RA-Semigroup And Conditions When Paramedial Groupoid S Becomes Commutative:
For readers to grasp the approach of section-4 without any difficulty we give refer- ences that if a groupoid S satisfies medial or paramedial law and contains identity element then groupoid S is commutative monoid proved in [11] and in [12].
Also we proved Theorem-19 and Theorem-20 in [14] that if S is medial, paramedial and satisfies left double displacement or right double displacement law then S is commutative. In the last section of this article we discussed that when a groupoid S satisfies paramedial law i.e. ∀a, b, c and d ∈ S the condition (ab)(cd) = (db)(ca) is satisfied then on what conditions S becomes LA-Semigroup, RA-Semigroup, commutative semigroup by following theorems:
Theorem 31: If S is semigroup, medial and paramedial then S is commuta- tive semigroup.
Proof: If S is paramedial and medial so∀a, b, c and d∈S the condition (ab)(cd) = (db)(ca) and (ab)(cd) = (ac)(bd) are satisfied and also (ab)(cd) = (dc)(ba). Also S is semigroup so (ab)c=a(bc).
From the theorem proved in [12]: if S is paramedial then S is medial if (a) S is unipotent and left(right) cancellative, (b) S contains left(right) neutral element (c) S is commutative, (d) S is left(right) modular or (e) S is idempotent. By using these five cases the proof of theorem needs discussion on all five cases.
We have other two cases which are following:
We proved Theorem-17 in [14] that if semigroup S satisfies the condition (ab)c= (ac)b then S is RDD-semigroup and we have proved this result that if S is medial, paramedial and RDD-semigroup then S is commutative in Theorem-20 [14]. We proved the Theorem-20 by the following steps:
(ab)(cd) = (db)(ca) = (da)(cb) = (dc)(ab) by using paramedial, RDD and me- dial law
(dc)(ab) = (bc)(ad) = (bd)(ac) = (ba)(dc) by using paramedial, RDD and medial law
(ba)(dc) = (ca)(db) = (cd)(ab) by using paramedial and medial law
We proved Theorem-18 in [14] that if S is semigroup and satisfies the condition that (ab)c=b(ac) = (ba)cthen S is medial and LDD-Semigroup. So if S is medial, paramedial, semigroup and LDD-semigroup then from theorem 20 in [14] this is effortless to prove that S is commutative. The theorem we proved has following steps:
(ab)(cd) = (db)(ca) = (cb)(da) = (cd)(ba) by using paramedial, LDD and me- dial law
(cd)(ba) = (ad)(bc) = (bd)(ac) = (ba)(dc) by using paramedial, LDD and medial law
(ba)(dc) = (ca)(db) = (cd)(ab) by using paramedial and medial law
Lemma-5: If S is paramedial and RA-Semigroup then ∀ a, b, c and d ∈ S, (ab)(cd) = (dc)(ba).
Proof: S is RA-Semigroup so surely S is medial proved and explained in [3] and [6] so we have following steps:
(ab)(cd) = (db)(ca) = (dc)(ba), by allying[] medial law.
Also we can use these steps that (ab)cd) = (db)(ca) = a(c(db)) = a(b(dc)) = (dc)(ba).
Semigroup, Medial And Paramedial Relation:
(a) This is not necessary that if S is medial and paramedial then S is commutative e.g. LA-Monoid is Paramedial and also RA-Monoid is paramedial but this is not necessary that if a groupoid S is LA-Monoid or RA-Monoid then S is commutative which are next in Theorems 32 and 33.
(b) This is not necessary that if S is semigroup and medial then S is commuta- tive, e.g. if binary operation on any non empty set is defined by such way that ab = b then S is medial, LDD-semigroup and semigroup and w.r.t binary oper- ation ab = a S is semigroup, medial and RDD-semigroup but S does not satisfy paramedial, commutative, left invertive and right invertive properties.
(c) This is also clear from the Theorems 1 to 7 that non commutative and non
associative groupoids can be medial and RDD-semigroup or medial and LDD- semigropup; see Answer to Open Problem in [14] and Recall Examples In [10] and Recall Examples In [14].
Theorem 32: If S is paramedial with right identity then S is RA-Monoid.
Proof: Given that ∀ a, b, c and d ∈ S, (ab)(cd) = (db)(ca), (ab)(cd) = (ac)(bd) because every paramedial is medial proved in [12]; and∃right identity e such that ae=a ∀ a∈ S.
a(bc) = (ae)(bc) = (ce)(ba) =c(ba) which shows that S is RA-Monoid and also S satisfies the condition (ab)c= (ab)(ce) = (ac)(be) = (ac)b.
Theorem 33: If S is RA-Monoid then S is paramedial.
Proof: Given that∀ a, b, c and d ∈S, (ab)(cd) = (ac)(bd) anda(bc) =c(ba) and also ∃ right identity in S such that ∀ a ∈S we have ae =a.
So (ab)c= (ab)(ce) = (ac)(be) = (ac)b. So (ab)(cd) = (ab)g = (ag)b = (a(cd))b = (d(ca))b.
Also (db)(ca) = (db)h = (dh)b = (d(ca))b. Thus (ab)(cd) = (db)(ca) = (d(ca))b and S is RA-monoid and paramedial. This is done by the indirect way.
Note in Paramedial Groupoids: Madad et al. [13]; proved that every LA- Monoid S satisfies paramedial law. We have the following theorem:
Theorem 34: If S is paramedial with left identity then S is LA-Monoid.
Proof: Given that ∀ a, b, c and d ∈ S, (ab)(cd) = (db)(ca), (ab)(cd) = (ac)(bd) because every paramedial groupoid is medial proved in [12]; and ∃ left identity e such that ea=a ∀ a∈ S.
So (ab)c= (ab)(ec) = (cb)(ea) = (cb)a which shows that S is LA-Semigroup.
Also a(bc) = (ea)(bc) = (eb)(ac) = b(ac).
Note in Theorem 34: Cho et al. [12] proved that if Paramedial groupoid X contains identity then X is commutative monoid. They used the method (ab)c = (ab)(ce) = (eb)(ca) = b(ca) and also (ab)c = (ab)(ec) = (cb)(ea) = (cb)a that makes X “right modular groupoid” i.e. LA-Semigroup with identity which is surely commutative monoid proved by kazim and naseeruddin [3].
Theorem 35: If S is medial, paramedial and satisfies the property (ab)c= (ba)c,
∀ a, b and c ∈ S then S is commutative.
Proof: Given that ∀ a, b, c and d ∈ S the conditions (ab)(cd) = (db)(ca) and (ab)c= (ba)c and (ab)(cd) = (ac)(bd) = (dc)(ba), so we have two following proce- dure to porove the required result:
(ab)(cd) = (ba)(cd) = (da)(cb) = (ad)(cb) = (bd)(ac), thus (ab)(cd) = (bd)(ac). So (ab)(cd) = (ac)(bd) = (cd)(ab) that shows that S is commutative.
Theorem 36: If S is medial, paramedial and ∀ a, b and c ∈ S holds one of the condition from
(a) (ab)c= (bc)a or (ab)c= (ca)b or (b) (ab)c= (ac)b, then S is commutative.
Proof (a): If∀a, b, c and d∈ groupoid S the condition (ab)c= (bc)ais satisfied then surely (ab)c= (ca)b so we have the following steps:
(ab)(cd) = ((cd)a)b = ((da)c)b = (cb)(da) = (cd)(ba) = (ad)(bc) = (ab)(dc) by applying the conditions, medial and paramedial laws.
So (ab)(cd) = (ad)(bc) = (ad)(cb) shows that S is RDD-semigroup and from Theorem-20 in [14] S is commutative so (ab)c= (bc)a=a(bc) so S is commutative semigroup. We have already explained Theorem-20 from [14] in Theorem-31.
Alternate Proof (a): (ab)(cd) = (ac)(bd) = (dc)(ba) = ((ba)d)c = ((ad)b)c = (bc)(ad) = (ba)(cd) = (da)(cb).
So (ab)(cd) = (ba)(cd) = (bc)(ad) = (cb)(ad) that shows that S is LDD-semigroup and we alreadcy proved Theorem-19 in [14] that if groupoid S is medial, parame- dial and LDD-semigroup then S is commutative.
Proof (b): To prove the second part of the theorem by direct way is not easy so we do substitution method. Clearly S satisfies the conditions (ab)(cd) = (db)(ca) = (dc)(ba) = (ac)(bd) and also this is given that (ab)c= (ac)b so we have the folow- ing steps:
(ab)(cd) = (a(cd))b and (ab)(cd) = (db)(ca) = (d(ca))b and (ab)(cd) = (ac)(bd) = (a(bd))c and (ab)(cd) = (dc)(ba) = (d(ba))c
So (ab)(cd) = (a(cd))b = (d(ca))b = (a(bd))c = (d(ba))c so we can apply these properties by the following step:
((ab)(cd))(f g) = ((a(cd))b)(f g) = ((f g)((cd)a))b= ((f g)(ab))(cd).
By applying (ab)(cd) = (a(cd))b, (ab)(cd) = (d(ba))c and (ab)(cd) = (a(cd))b So this is generally proved that∀a, b, c, d, e and f∈S the condition ((ab)(cd))(f g) = ((f g)(ab))(cd) so if binary operation is well defined then surely if a and b∈S then ab ∈ S also. Let ab = x, cd = y and f g = z then ∀ x, y and z ∈ S we have (xy)z = (zx)y, (xy)z = (yz)x and (xy)z = (yx)z with the conditions that S is medial and paramedial so clearly S is commutative by using Theorem-35 and Theorem-36(a) and therefore S is commutative semigroup.
Theorem 37: If groupoid S is medial, satisfies the condition (ab)c = (ac)b ∀ a, b, c and d∈ S and contains left identity then S is commutative monoid.
Proof: Using the given conditions (ab)(cd) = (a(cd))b = ((ea)(cd))b= ((ec)(ad))b= ((ec)b)(ad)) = (cb)(ad) which shows that S is LDD-semigroup.
Now (ab)c= (ab)(ec) = (eb)(ac) = b(ac) and so we can say if (ab)c= b(ac) then (ab)c= (ac)b=c(ab).
So (ab)c = (ac)b =b(ac) = c(ab). Therefore c(ab) = (ec)(ab) = (ea)(cb) = a(cb).
Thus (ab)c= (ac)b = a(cb) and therefore S satsifies all these conditions (ab)c = (ac)b =b(ac) =a(cb) = c(ab)
Clearly (ab)c = b(ac) = (eb)(ac) = (ea)(bc) by medial law implies that (ab)c = b(ac) = a(bc) so S is semigroup.
So S is semigroup, medial, LDD-semigroup and contains left identity and satisfies the conditions (ab)c = a(bc) and (ab)c = b(ac) = (ba)c, (ab)c = (ac)b = a(cb),
(ab)c=c(ab) = (ca)band also (ab)c=a(bc) = (ba)c=c(ba) = (cb)aso therefore S is LA-Semigroup as well as RA-Semigroup therefore from the Results-1 and from [3] and [6] S is commutative monoid.
Remarks Related to Theorem 37:
(a) If a groupoid S is medial and contains left identity then a(bc) = (ea)(bc) = (eb)(ac) =b(ac).
(b) If a groupoid S is medial and contains right identity then (ab)c= (ab)(ce) = (ac)(be) = (ac)b.
Theorem 38: If S is paramedial, contains left idenitity and satisfies condition (ab)c=b(ac) then S is commutative monoid.
Proof: S is paramedial with left identity then S is surely LA-Monoid by theorem- 34.
So (ab)c=b(ac) = (eb)(ac) = ((ac)b)e= ((bc)a)e= (ea)(bc) =a(bc) which shows that S is semigroup and this is in Proved Results-1 that if S is LA-semigroup and associative then S is commutative semigroup.
Theorem 39: If S is medial, satisfies the condition (ab)c = (ba)c, ∀ a, b, c and d ∈ S and contains right identity then S is commutative monoid.
Proof: Given that∀ a, b, c and d ∈S the consitions (ab)(cd) = (ac)(bd), (ab)c= (ba)cand∃right identity e such thatae=aso (ab)c= (ab)(ce) = (ac)(be) = (ac)b that shows that (ab)c= (ac)b = (ca)b.
So (ab)c= (ca)b = (ca)(be) = (cb)(ae) = (cb)a so S is LA-Semigroup with right identity and this is proved in [3] and in [6] that LA-semigroup with right identity is commutative monoid.
Theorem 40: Without using right or left identity tool prove that if a groupoid S is paramedial and almost semigroup i.e. S satisfies paramedial law both left invertive and right invertive laws then S is commutative semigroup.
Proof: Given that S satisfies left invertive law, right invertive law and parame- dial law so surely S satisfies medial law from [3] and [12] so ∀ a, b, c and d ∈ S we have (ab)(cd) = (ac)(bd) = (db)(ca) = (dc)(ba), and also (ab)c = (cb)a and a(bc) = c(ba). There is no easy direct way so we take five elements and prove the thorem by the follwoing steps:
(ab)((cd)f) = (ab)((f d)c) =c((f d)(ab)) = c((bd))(af)).
Soc((bd)(af)) =c((ba)(df)) = (df)((ba)c) = (df)((ca)b).
So (df)((ca)b) =b((ca)(df)) = b((cd)(af)) = (af)((cd)b).
From the steps this is clear that ∀ a, b, x and f ∈ S, (ab)(xf) = (af)(xb) so that shows that S is RDD-semigroup and we can use Theorem-20 from [14] that if S is medial, paramedial and RDD-semigroup then S is commutative. Also S is surely a groupoid and if binary operation is well defined then ∀a and b ∈S ab ∈ S also. So if c and d ∈S then cd ∈ S also.
Next Work:
(a): Finite Groupoids Pattern Containing Bands Which Are LDD-semigroups, RDD-semigroups And Contatning LA-Semigroups, RA-Semigroups With Cyclic Groups.
(b): Survey On Semigroup Behaviour When Semigroup Becomes Medial, Para- medial, RDD-semigroup and LDD-semigroup.
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