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 Authors : Vb Subrahmanyeswara Rao Seetamraju, A. Anjaneyulu, D. Madhusudana Rao
 Paper ID : IJERTV1IS6525
 Volume & Issue : Volume 01, Issue 06 (August 2012)
 Published (First Online): 02092012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Partially Ordered Γ Semigroups
VB Subrahmanyeswara Rao Seetamraju1, A. Anjaneyulu2, D. Madhusudana Rao3.
1Dept. of Mathematics, V K R, V N B & A G K College Of Engineering, Gudivada, A.P. India.
2Dept. of Mathematics, V S R & N V R College, Tenali, A.P. India.
3Dept. of Mathematics, V S R & N V R College, Tenali, A.P. India.
Abstract
In this paper, the notion of an ordered semigroup is introduced and some examples are given. Further the terms commutative ordered semigroup, quasi commutative ordered semigroup, normal ordered – semigroup, left pseudo commutative ordered – semigroup, right pseudo commutative ordered – semigroup are introduced. It is proved that (1) if S is a commutative ordered semigroup then S is a quasi commutative ordered semigroup, (2) if S is a quasi commutative ordered semigroup then S is a normal ordered semigroup, (3) if S is a commutative ordered semigroup, then S is both a left pseudo commutative and a right pseudo commutative ordered semigroup. Further the terms; left identity, right identity, identity, left zero, right zero, zero of an ordered semigroup are introduced. It is proved that if a is a left identity and b is a right identity of an ordered semigroup S, then a = b. It is also proved that any ordered – semigroup S has at most one identity. It is proved that if a is a left zero and b is a right zero of an ordered – semigroup S, then a = b and it is also proved that any ordered semigroup S has at most one zero element. The terms; ordered subsemigroup, ordered – subsemigroup generated by a subset, idempotent, idempotent, strongly idempotent, midunit, relement, regular element, left regular element, right regular element, completely regular element, (, )inverse of an element, semisimple element and intra regular element in an ordered semigroup are introduced. Further the terms idempotent ordered semigroup and generalized commutative ordered semigroup are introduced. It is proved that every idempotent element of an ordered semigroup is regular. It is also proved that, in an ordered semigroup, a is a regular element if and only if a has an ( , )inverse. It is proved that, (1) if a is a completely regular element of
an ordered semigroup S, then a is both left regular and right regular, (2) if a is a completely regular element of an ordered semigroup S, then a is regular and semisimple, (3) if a is a left regular element of an ordered semigroup S, then a is semisimple, (4) if a is a right regular element of an ordered semigroup S, then a is semisimple, (5) if a is a regular element of an ordered – semigroup S, then a is semisimple and
(6) if a is a intra regular element of an ordered – semigroup S, then a is semisimple. The term separative ordered
semigroup is introduced and it is proved that, in a separative ordered semigroup S, for any x, y, a, b S, the statements (i) xa xb if and only if ax bx, (ii) x xa xxb implies xa xb,(iii) xya
xyb implies yxa yxb hold.

Introduction
– semigroup was introduced by Sen and Saha [15] as a generalization of semigroup. Anjaneyulu. A [1], [2] and [3] initiated the study of ideals and radicals in semigroups. Many classical notions of semigroups heve been extended to semigroups to semigroups by Madhusudhana Rao, Anjaneyulu and Gangadhara Rao [11]. The concept of partially ordered semigroup was introduced by Y. I. Kwon and S. K. Lee [10] in 1996, and it has been studied by several authors. In this paper we introduce the notions of ordered – semigroups and characterize ordered semigroups.

PRELIMINARIES :
DEFINITION 2.1: Let S and be any two nonempty sets. S is called a semigroup if there exist a mapping from S S to S which maps (a, , b) a b satisfying the condition : (ab)c = a(bc) for all a,b,c
M and , .
NOTE 2.2: Let S be a semigroup. If A and B are subsets of S, we shall denote the set
{ ab : aA , bB and } by AB.
DEFINITION 2.3: A semigroup S is said to be commutative provided ab = ba for all a,b S and .
NOTE 2.4: If S is a commutative semigroup, then
a b = b a for all a, b S.
DEFINITION 2.5 : A semigroup S is said to be quasi commutative provided for each a,b S, there exists a natural number n such that
DEFINITION 2.13 : A semigroup S is said to be right pseudo commutative provided abc = acb for all a,b,c S.
THEOREM 2.14 : If S is a commutative – semigroup, then S is both a left pseudo commutative
semigroup and a right pseudo commutative – semigroup.
DEFINITION 2.15 : An element a of a semigroup S is said to be a left identity of S provided as = s for all s S and .
DEFINITION 2.16 : An element a of a semigroup S is said to be a right identity of S provided sa = s for all s S and .
DEFINITION 2.17 : An element a of a semigroup S is said to be a two sided identity or an identity provided it is both a left identity and a right identity of S.
THEOREM 2.18 : If a is a left identity and b is a
a b (b
)n a .
right identity of a semigroup S,
then a = b.
NOTE 2.6 : If a semigroup S is quasi commutative then for each a,b S , there exists a natural number n such that, ab = (b )na.
THEOREM 2.7: If S is a commutative semigroup then S is a quasi commutative semigroup.
DEFINITION 2.8 : A semigroup S is said to be
normal provided
DEFINITION 2.19 : An element a of a semigroup S is said to be a left zero of S provided as = a for all s S and .
DEFINITION 2.20 : An element a of a semigroup S is said to be a right zero of S provided sa = a for all s
S and .
DEFINITION 2.21 : An element a of a semigroup S is said to be a two sided zero or zero provided it is both
a S S a
and
a S .
a left zero and a right zero of S.
NOTE 2.9 : If a semigroup S is normal then aS =
Sa for all a S.
THEOREM 2.10 : If S is a quasi commutative – semigroup then S is a normal semigroup.
COROLLARY 2.11 : Every commutative – semigroup is a normal semigroup.
DEFINITION 2.12 : A semigroup S is said to be left pseudo commutative provided abc = bac for all a,b,c S.
DEFINITION 2.22: An element a of – semigroup S is said to be a – idempotent provided aa = a for all .
NOTE 2.23: If an element a of – semigroup S is a – idempotent, then aa = a.
DEFINITION 2.24: A – Semigroup S is said to be an idempotent semigroup or a band provided every element in S is a – idempotent.
DEFINITION 2.25 : An element a of semigroup S is said to be a midunit provided xay = xy for all x, y S.
NOTE 2.26 : Identity of a – semigroup S is a midunit.
DEFINITION 2.27 : An element a of – semigroup S is said to be an relement provided as = sa for all s S and if x, y S, then axy = byx for some b S.
DEFINITION 2.28 : A – semigroup S with identity 1 is said to be a generalized commutative – semigroup provided 1 is an relement in S.
DEFINITION 2.29 : An element a of a semigroup S is said to be regular provided a = a x a, for some x S and , . i.e, a aSa.
DEFINITION 2.30 : A ideal A of a semigroup S is said to be regular if every element of A is regular in A.
DEFINITION 2.31 : A – semigroup S is said to be a regular – semigroup provided every element is regular.
DEFINITION 2.32 : An element a of a semigroup S is said to be left regular proided a = a a x, for some x S and , . i.e, a aaS.
DEFINITION 2.33 : An element a of a semigroup S is said to be right regular provided a = x a a, for some x S and , . i.e, a Saa.
DEFINITION 2.34 : An element a of a semigroup S is said to be completely regular provided, there exists an element x S such that a = a x a for some , and a x = a i.e., a axa and ax = xa.
DEFINITION 2.35 : A semigroup S is said to be completely regular semigroup provided every element of S is completely regular.
DEFINITION 2.36 : An element a of – semigroup S is said to be semisimple provided a < a > < a >, that is, < a > < a > = < a >.
DEFINITION 2.37 : A – semigroup S is said to be semisimple – semigroup provided every element of S is a semisimple element.
DEFINITION 2.38 : An element a of a semigroup S is said to be intra regular provided a = xaay for some x, y S and , , .
THEOREM 2.39 : If a is a completely regular element of a – semigroup S, then a is regular and semisimple.
THEOREM 2.40 : If a is a intra regular element of a – semigroup S, then a is semisimple.

ORDERED SEMIGROUP :
DEFINITION 3.1 : A semigroup S is said to a partially ordered semigroup if S is partially ordered set such that a b a c b c and
c a c b a,b, c S and .
NOTE 3.2 : A partially ordered semigroup simply called po–semigroup or ordered semigroup.
EXAMPLE 3.3 : Let S = { , {a}, {b}, {c}, {a, b},
{b, c}, {a, c}, {a, b, c}} and = {, {a}, {a, b, c}}. If for all A, C S and B , ABC = A B C and
A C A C, then S is an ordered semigroup.
EXAMPLE 3.4: 1.1.8 : Let S = {, {a}, {b}, {c},
{a, b}, {b, c}, {a, c}, {a, b, c}} and = {{a, b, c}}. If for all A, C S and B , ABC = A B C and A C A C, then S is an ordered semigroup.
NOTE 3.5: In general, let P(X) be the power set of any nonempty set X and a topology on X. If we define for all A, C S and B , ABC = A B C and A C A C, then P(X) is an ordered semigroup.
EXAMPLE: 3.6 : Let S be the set of all 2 Ã— 3 matrices over Q, the set of positive integers and be the set of all 3 Ã— 2 matrices over the same set. Define AB = usual matrix product of A, , B; for all A, B S and for all . Then S is a semigroup. Note that S is not a semigroup. Also S and are posets with respect to defined by (aik) (bik) if and only if aik bik for all i, k. Then S is an ordered semigroup.
EXAMPLE: 3.7 : Let S be the set of all integers of the form 4n+1 where n is an integer and denote the set of all integers of the form 4n+3. If ab is a++b (usual sum of the integers) and a b means a is less than or equal to b, for all a, b S and , then S is an ordered semigroup.
EXAMPLE 3.8 : Let M be the set of all isotone each a,b S, there exists a natural number n such that
mappings from a poset P into another poset Q and the set of all isotone mappings from the poset Q into the poset P. Let f , g M and . Define by fg, the usual mapping composition of f, and g. Then M is a
semigroup. We define a relation on M by f g if and only if af ag, for all a P. This relation is a partial order on M and as such M is a poset. We also define a relation on by if and only if x x
, for all x Q. For this relation is a poset. Then M is an ordered semigroup.
EXAMPLE 3.9 : For a, b [0,1], let M = [0, a] and = [0, b]. Then M is an ordered semigroup under usual multiplication and usual partial order relation.
EXAMPLE 3.10 : Fix m Z, and let M be the set of all integers of the form mn +1 and denotes the set of all integers of the form mn + m 1 where n is an integer. Then M is a ordered semigroup under usual addition and usual partial order relation.
In the following we introduce the notion of a commutative ordered semigroup.
DEFINITION 3.11 : An ordered semigroup S is said to be ordered commutative semigroup
a b (b )n a .
NOTE 3.17 : If an ordered semigroup S is ordered quasi commutative semigroup then for each a,b S , there exists a natural number n such that, ab = (b )na.
THEOREM 3.18 : If S is an ordered commutative semigroup then S is an ordered quasi commutative semigroup.
Proof: Suppose that S is an ordered commutative – semigroup.
Then S commutative semigroup. Let a, b S.
Now a b b a a b (b )1 a . Therefore S is a quasi commutative semigroup.
Therefore S is an ordered quasi commutative –
semigroup.
In the following we introduce the notion of an ordered normal semigroup.
DEFINITION 3.19 : An ordered semigroup S is said to be ordered normal semigroup provided S is normal semigroup.
NOTE 3.20 : An ordered semigroup S is said to be ordered normal semigroup provided
provided S is a commutative semigroup.
a S S a
and
a S .
NOTE 3.12 : An ordered semigroup S is said to be ordered commutative semigroup provided ab = ba for all a,b S and .
NOTE 3.13 : If S is an ordered commutative – semigroup then ab = ba for all a, b S.
NOTE 3.14 : Let S be a semigroup and a, b S and . Then aab is denoted by (a)2b and consequently a a a ..(n terms)b is denoted by (a)nb.
In the following we introduce a quasi commutative ordered semigroup.
DEFINITION 3.15 : An ordered semigroup S is said to be ordered quasi commutative semigroup provided S is a quasi commutative – semigroup.
NOTE 3.16 : An ordered semigroup S is said to be ordered quasi commutative semigroup provided for
NOTE 3.21 : If an ordered semigroup S is ordered normal semigroup then aS = Sa for all a S.
THEOREM 3.22 : If S is an ordered quasi commutative semigroup then S is a normal ordered semigroup.
Proof: Suppose that S is an ordered quasi commutative
– semigroup.
Then S is quasi commutative semigroup. Let a S and
Let x a S. Then x = a b, where b S
Since S is quasi commutative, a b = (b)na for some
n N
Therefore, x = a b = (b)na = (b)n1ba S a a S S a —(1)
Let x S a. Then x = t a, for some t S
Since S is quasi commutative, t a = (a)nt for some
n N
Now, x = t a = (a)nt = a (a)n1t a S S a a S —(2)
From (1) and (2), a S = S a for all and for all b S
Hence S is a normal semigroup. Therefore S is an ordered normal semigroup.
COROLLARY 3.23 : Every ordered commutative
semigroup is a normal ordered
semigroup.
Proof : Let S be an ordered commutative semigroup. By theorem 3.18, S is an ordered quasi commutative – semigroup. By theorem 3.22, S is an ordered normal semigroup. Therefore every ordered commutative semigroup is an ordered normal semigroup.
In the following we are introducing ordered left pseudo commutative semigroup and ordered right pseudo commutative semigroup.
DEFINITION 3.24 : An ordered semigroup S is said to be ordered left pseudo commutative – semigroup provided S is left pseudo commutative – semigroup.
NOTE 3.25: An ordered semigroup S is said to be ordered left pseudo commutative provided abc = bac for all a,b,c S.
DEFINITION 3.26 : An ordered semigroup S is said to be ordered right pseudo commutative – semigroup provided S is right pseudo commutative – semigroup.
NOTE 3.27 : An ordered semigroup S is said to be ordered right pseudo commutative semigroup provided abc = acb for all a,b,c S.
THEOREM 3.28 : If S is an ordered commutative semigroup, then S is both an ordered left pseudo commutative semigroup and an ordered right pseudo commutative semigroup.
Proof : Suppose that S is ordered commutative – semigroup.
Then S is commutative semigroup.
Then abc = (ab)c= (ba)c = bac./p>
Therefore S is left pseudo commutative semigroup. Therefore S is an ordered left pseudo commutative – semigroup.
Again abc = a(bc) = a(cb) = acb.
Therefore S is a right pseudo commutative – semigroup.
Therefore S is an ordered right pseudo commutative – semigroup.
NOTE 3.29 : The converse of the above theorem is not true. i.e., if S is an ordered left and right pseudo commutative semigroup then S need not be an ordered commutative
semigroup.
EXAMPLE 3.30 : Let S = {a , b , c }and = { x, y, z } Define a binary operation . in S as shown in the following table:
.
a
b
c
a
a
a
a
b
a
a
a
c
a
b
c
Define a mapping S Ã— Ã— S S by ab = a.b for all a, b S and . It is easy to see that S is a ordered semigroup. Now S is an ordered left and right pseudo commutative semigroup. But S is not an ordered commutative – semigroup.
In the following we are introducing left identity, right identity and identity of a ordered – semigroup.
DEFINITION 3.31 : An element a of an ordered – semigroup S is said to be a left identity of S provided as = s and s a for all s S and .
DEFINITION 3.32 : An element a of an ordered – semigroup S is said to be a right identity of S provided sa = s and s a for all s S and .
DEFINITION 3.40 : An element a of an ordered – semigroup S is said to be a two sided identity or an identity provided it is both a left identity and a right identity of S.
NOTE 3.41 : An element a of an ordered – semigroup S is said to be a two sided identity or an identity provided sa = as = s and s a for all s S and .
EXAMPLE 3.42 : 1.1.7 : Let S = { , {a}, {b}, {c},
{a, b}, {b, c}, {a, c}, {a, b, c}} and = {}. If for all A, C S and B , ABC = A B C and A C A C, then S is a partially ordered – semigroup with identity.
EXAMPLE: 3.43 : Let S = {, {a}, {b}, {c}, {a, b},
{b, c}, {a, c}, {a, b, c}} and = {{a, b, c}}. If for all A, C S and B , ABC = A B C and
A C A C, then S is a partially ordered – semigroup with identity.
THEOREM 3.44 : If a is a left identity element and b is a right identity element of an ordered – semigroup S, then a = b.
Proof: Since a is a left identity of S, a s s and s
a for all s S and and hence a b = b and b
a for all . Since b is a right identity of S, s b s and s b for all s S and and hence a b = a and a b for all . Now b a and a b
a = b.
THEOREM 3.45 : Any ordered semigroup S has at most one identity.
Proof : Let a, b be two identity elements of the ordered semigroup S. Now a can be considered as a left identity and b can be considered as a right identity of S. By theorem 3.44, a = b. Then S has at most one identity.
NOTE 3.46 : The identity (if exists) of an ordered – semigroup is usually denoted by 1 or e.
DEFINITION 3.47 : A partially ordered semigroup S with identity is called a partially ordered monoid.
In the following we are introducing left zero, right zero and zero of an ordered semigroup.
DEFINITION 3.48: An element a of an ordered –
semigroup S is said to be a left zero of S provided as
= a and a s for all s S and .
DEFINITION 1.1.36 : An element a of an ordered – semigroup S is said to be a right zero of S provided sa = a and a s for all s S and .
DEFINITION 3.49 : An element a of an ordered – semigroup S is said to be a two sided zero or zero provided it is both a left zero and a right zero of S.
NOTE 3.50 : An element a of an ordered semigroup S is said to be a two sided zero or zero provided as = sa = a and a s for all s S and .
EXAMPLE3.51 : Let S = {, {a}, {b}, {c}, {a, b},
{b, c}, {a, c}, {a, b, c}} and = {{a, b, c}, , {a}}. If for all A, C S and B , ABC = A B C and
A C A C, then S is a partially ordered – semigroup with zero.
EXAMPLE 3.52 : Let S = {, {a}, {b}, {c}, {a, b},
{b, c}, {a, c}, {a, b, c}} and = {{a, b, c}}. If for all A, C S and B , ABC = A B C andA C A C, then S is a partially ordered semigroup with zero and identity.
NOTE 3.53 : In general, let P(X) be the power set of any nonempty set X and a topology on X. If we define for all A, C S and B , ABC = A B C andA C A C, then P(X) is a partially ordered – semigroup with zero.
We are now introduce left zero ordered semigroup, right zero ordered semigroup and zero ordered – semigroup.
DEFINITION 3.54 : An ordered semigroup in which every element is a left zero is called a left zero ordered semigroup.
DEFINITION 3.55 : An ordered semigroup in which every element is a right zero is called a right zero ordered semigroup.
DEFINITION 3.56 : An ordered semigroup S with 0 in which the product of any two elements equals to 0 and 0 s for all s S is called a zero ordered – semigroup or a null ordered semigroup.
THEOREM 3.57 : If a is a left zero element and b is a right zero element of an ordered semigroup S, then a = b.
Proof: Since a is a left zero of S, a s a and a s for all s S, and hence a b =a and a b for all . Since b is a right zero of S, s b b and b
s for all s S and and hence a b = b and b
a for all . Now a b and b a a = b.
THEOREM 3.58 : Any ordered semigroup S has at most one zero element.
Proof : Let a, b be two zeros of the semigroup S. Now a can be considered as a left zero and b can be considered as a right zero. By theorem 3.57, a = b. Thus S has at most one zero.
NOTE 3.59 : The zero (if exists) of an ordered – semigroup is usually denoted by 0.
NOTATION 3.60 : Let S be an ordered semigroup. If S has an identity, let S1 = S and if S does not have an identity, let S1 be the ordered semigroup S with an identity adjoined usually denoted by the symbol 1. Similarly if S has a zero, let S0 = S and if S does not have a zero, let S0 be the ordered semigroup S with zero adjoined usually denoted by the symbol 0.
NOTATION 3.61 : Let S be an ordered semigroup and T is a nonempty subset of S. If H is a nonempty subset of T, we denote {t T : t h for some h H} by (H]T.
{t T : h t for some h H} by [H)T. (H]s and [H)s are simply denoted by (H] and [H) respectively.
THEOREM 3.62 : Let S be an ordered semigroup and A S, B S. Then (i) A (A], (ii) ((A]] = (A], (iii) (A] (B] (A B]
and (iv) A (B] for A B,
(v) (A] (B] for A B .
Proof : (i) Let x A. x A x S and x x x
(A]. Therefore A (A].

Let x ((A]] x y for some y (A] y z for some z A. x y, y z x z . Thus x (A]. Therefore ((A]] (A] and from (i) A (A] (A] = ((A]] and hence ((A]] = (A].

Let x (A] (B] x a b for some a A, , b B x a b for some a b AB x (AB]. Therefore (A] (B] (AB].

From (i) B (B] A B (B].

A B A (B] (A] ((B]] = (B] Therefore (A] (B].
DEFINITION 3.63 : Let S be an ordered – semigroup. A nonempty subset T of S is said to be an
ordered subsemigroup of S if ab T, for all a, b T and and t T, s S, s t s T.
EXAMPLE 3.64 : Let S = [ 0,1] and = { 1/n : n is a positive integer}. Then S is an ordered semigroup under the usual multiplication and usual order relation. Let T = [0, 1/2]. Now T is a nonempty subset of S and ab T, for all a, b T and . Then T is an ordered subsemigroup of S.
THEOREM 3.65 : A nonempty subset T of an ordered semigroup S is an ordered subsemigroup of S if (1) TT T, (2) (T] T.
THEOREM 3.66 : The nonempty intersection of two ordered subsemigroups of an ordered – semigroup S is an ordered subsemigroup of S.
Proof : Let T1, T2 be two ordered subsemigroups of
S. Let a, b T1T2 and .
a, b T1T2 a, b T1 and a, b T2
a, b T1, , T1 is an ordered subsemigroup of S
a b T1 and (T1] T1.
a, b T2, , T2 is a subsemigroup of S a b
T2 and (T2] T2.
a b T1, a b T2 a b T1T2 and T1T2 T1, T1T2 T2
(T1T2] (T1] = T1 and (T1T2] (T2] = T2
(T1T2] T1T2 (T1T2] = T1T2. and hence T1T2 is an ordered subsemigroup of S.
THEOREM 3.67 : The nonempty intersection of any family of ordered
subsemigroups of an ordered semigroup S is an ordered subsemigroup of S.
In the following we are introducing an ordered subsemigroup which is generated by a subset and a cyclic ordered subsemigroup of an ordered – semigroup.
DEFINITION 3.68 : Let S be an ordered semigroup and A be a nonempty subset of S. The smallest ordered subsemigroup of S containing A is called an ordered
subsemigroup of S generated by A. It is denoted by
< A >.
THEOREM 3.69 : Let S be an ordered semigroup and A be a nonempty subset of S. Then < A > = the
intersection of all ordered subsemigroups of S containing A.
DEFINITION 3.70 : Let S be an ordered – semigroup. An ordered subsemigroup T of S is said to be cyclic ordered subsemigroup of S if T is generated by a single element subset of S.
DEFINITION 3.71 : An ordered semigroup S is said to be a cyclic ordered semigroup if S is a cyclic ordered subsemigroup of S itself.


SPECIAL ELEMENTS OF AN ORDERED – SEMIGROUP
We now introduce idempotent element and idempotent element in an ordered semigroup.
DEFINITION 4.1 : An element a of an ordered – semigroup S is said to be a
idempotent provided a aa.
NOTE 4.2 : The set of all idempotent elements in an ordered – semigroup S is denoted by E S .
THEOREM 4.3 : For any ordered semigroup S,
E S with the binary relation defined by
a b iff a = a b = b a is a patially ordered set.
DEFINITION 4.4 : An element a of an ordered – semigroup S is said to be an idempotent or – idempotent if a aa for all .
NOTE 4.5: An element a of an ordered – semigroup S is said to be an idempotent or idempotent if a (aa]
NOTE 4.6 : In an ordered semigroup S, a is an idempotent iff a is an idempotent for all .
We now introduce an idempotent ordered – semigroup and a strongly idempotent ordered – semigroup.
DEFINITION 4.7 : An ordered semigroup S is said to be an ordered idempotent semigroup provided every element of S is idempotent for some .
DEFINITION 4.8 : An ordered semigroup S is said to be an ordered strongly idempotent – semigroup provided every element in S is an idempotent.
We now introduce a special element which is known as midunit in an ordered semigroup.
DEFINITION 4.9 : An element a of an ordered – semigroup S is said to be a midunit provided (xay]
= (xy] for all x, y S.
NOTE 4.10 : Identity of an ordered – semigroup S is a midunit.
We now introduce an relement in an ordered semigroup and also a generalized commutative ordered semigroup.
DEFINITION 4.11 : An element a of ordered – semigroup S is said to be an relement provided (as] =(sa] for all s S and if x, y S, then (axy] = (byx] for some b S.
DEFINITION 4.12 : An ordered – semigroup S with identity 1 is said to be an ordered generalized commutative – semigroup provided 1 is an relement in S.
We now introduce a regular element in an ordered semigroup and regular ordered semigroup.
DEFINITION 4.13 : An element a of an ordered – semigroup S is said to be regular provided a a x a, for some x S and , .
NOTE 4.14 : An element a of an ordered semigroup S is said to be regular provided a (aSa].
DEFINITION 4.15 : An ordered – semigroup S is said to be an ordered regular – semigroup provided every element is regular.
. 
0 
a 
b 
0 
0 
0 
0 
a 
0 
a 
a 
b 
0 
b 
b 
EXAMPLE 4.16: Let S = {0, a, b} and be any nonempty set. If we define a binary operation on S as the following Cayley table, then S is a semigroup.
Define a mapping from S Ã— Ã— S to S as a b = ab for all a, b S and . Then S is ordered regular – semigroup under usual partial order relation.
THEOREM 4.17 : Every idempotent element in an ordered semigroup is regular
Proof : Let a be an idempotent element in an ordered semigroup S.
then a aa for some . Hence a aa a. Therefore a is a regular element.
We now introduce left regular element, right regular element, completely regular element in an ordered semigroup and completely regular ordered – semigroup.
DEFINITION 4.18 : An element a of an ordered – semigroup S is said to be left regular provided a a a x, for some x S and , . i.e, a (aaS].
DEFINITION 4.19 : An element a of an ordered – semigroup S is said to be right regular provided a x a a, for some x S and , . i.e, a (Saa].
DEFINITION 4.20 : An element a of an ordered – semigroup S is said to be completely regular provided, there exists an element x S such that a a x a for some , and a x = a.
NOTE 4.21 : An element a of an ordered semigroup S is said to be completely regular provided, there exists an element x S such that a (axa ] and ax
= xa.
DEFINITION 4.22 : An ordered semigroup S is said to be ordered completely regular semigroup provided every element of S is completely regular.
DEFINITION 4.23 : Let S be an ordered – semigroup, a S and , . An element b S is said to be an ( , )inverse of a if a a b a and b a b.
THEOREM 4.24 : Let S be an ordered semigroup and a S. If a has an ( , )inverse then a is a regular element.
Proof : Suppose that b is an ( , )inverse of a.
Then a a b a and b a b. Therefore a
a b a and hence a is regular.
We now introduce a semisimple element of a semigroup and a semisimple semigroup.
DEFINITION 4.25 : An element a of an ordered – semigroup S is said to be semisimple provided a (< a
> < a >], that is, (< a >] = ( < a > < a >].
DEFINITION 4.26 : A – semigroup S is said to be ordered semisimple – semigroup provided every element of S is a semisimple element.
We now introduce an intra regular element of an ordered semigroup.
DEFINITION 4.27 : An element a of a semigroup S is said to be intra regular provided a xaay for some x, y S and , , .
NOTE 4.28 : An element a of a semigroup S is said to be intra regular provided a (SaaS].
EXAMPLE 4.29 : The semigroup given in example

is an intra regular ordered
semigroup.
THEOREM 4.30: If a is a completely regular element of an ordered – semigroup S, then a is regular and semisimple.
Proof : Since a is a completely regular element in the – smigroup S, a = a x a for some , and x S. Therefore a is regular.
Now a a x a axa < a > < a > a (< a >
< a >]. Therefore a is semisimple.
THEOREM 4.31 : If a is a completely regular element of an ordered – semigroup S, then a is both a left regular element and a right regular element.
Proof : Suppose that a is completely regular. Then a
(aSa] and aS = Sa.
Now a (aSa] = (aaS]. Thereforea is left regular.
Also a (aSa] = (Saa]. Therefore a is right regular.
THEOREM 4.32 : If a is a left regular element of a semigroup S, then a is semisimple.
Proof: Suppose that a is left regular. Then a (aax] and hence a (< a > < a >]. Therefore a is semisimple.
THEOREM 4.33 : If a is a right regular element of a semigroup S, then a is semisimple.
Proof: Suppose that a is right regular. Then a (aax] and hence a (< a > < a >]. Therefore a is semisimple.
THEOREM 4.34 : If a is a regular element of a – semigroup S, then a is semisimple.
Proof: Suppose that a is regular element of – semigroup S.
Then a a x a, for some x S, , a
(axa] and hence a (< a > < a >]. Therefore a is semisimple.
THEOREM 4.35 : If a is a intra regular element of a – semigroup S, then a is semisimple.
Proof: Suppose that a is intra regular. Then a (xaay] for x, y S and hence a (< a > < a >] Therefore a is semisimple.
DEFINITION 4.36 : An element a of an ordered – semigroup S is said to be left cancellative provided for , ab ac implies b c .
DEFINITION 4.37 : An element a of an ordered – semigroup S is said to be right cancellative provided for , ba ca implies b c.
DEFINITION 4.38 : An element a of an ordered – semigroup S is said to be cancellative provided a is both a left cancellative element and a right cancellative element.
DEFINITION 4.39 : An element a of an ordered – semigroup S is said to be left cancellative provided a is left cancellative for all
.
DEFINITION 4.40 : An element a of an ordered – semigroup S is said to be right cancellative provided a is right cancellative for all .
DEFINITION 4.41 : An element a of an ordered – semigroup S is said to be cancellative provided a is a both left cancellative and right cancellative.
NOTE 4.42 : ab cd if and only if ab cd.
DEFINITION 4.43 : An element a of an ordered – semigroup S is said to be strongly left cancellative provided a b a c implies b c .
NOTE 4.44 : An element a of an ordered – semigroup S is said to be strongly left
cancellative provided a b c , , b c.
DEFINITION 4.45 : An element a of an ordered – semigroup S is said to be strongly right cancellative provided b a c a implies b c .
NOTE 4.46 : An element a of an ordered – semigroup S is said to be strongly right
cancellative provided b a a , , b c.
DEFINITION 4.47 : An element a of an ordered – semigroup S is said to be strongly
cancellative provided a is a both strongly left – cancellative and strongly right
cancellative.
Now we introduce a weakened version of strongly cancellative ordered semigroup is provided by the following notion.
DEFINITION 4.48 : An ordered semigroup S is said to be separative if for any x, y S,

xx xy and yy yx imply x = y,

xx yx and yy xy imply x = y.

THEOREM 4.52 : In a separative ordered – semigroup S, for any x, y, a, b S, the following statements hold.

x a x b if and only if a x b x,

x x a x x b implies x a x b,

x y a x y b implies y x a y x b.
Proof : (i) If xa xb, then a(xa)x a(xb)x and b(xa)x b(xb)x so that (ax)(ax) (ax)(bx) and (bx)(ax) (bx)(bx) which by separativity implies (ax) (bx). The opposite implication follows by symmetry.

If xxa xxb, then by part (i), xax xbx
(ax)(ax) (ax)(bx) and (bx)(ax) (bx)(bx) and thus by separativity (ax) (bx). But then by part (i) xa xb.

Let xya xyb. Then xyay xyby, and thus by part (i), yayx ybyx. Multiplying by suitable elements on the right and using part (i), we obtain the following strings of inequalities:
yayxa ybyxa yayxb ybyxb ayxay byxay ayxby byxby (ay x)(ay x) (by x)(ay x)
(ay x)(by x) (by x)(by x)
Which by separativity implies (ay x) (by x).
But then by part (i), we have
yxa yxb.
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