Partially Ordered Γ- Semigroups

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, r-element, 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.

1. 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.

2. PRELIMINARIES :

   DEFINITION 2.1: Let S and be any two non-empty 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 r-element 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 r-element 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.

3. 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 × 3 matrices over Q, the set of positive integers and be the set of all 3 × 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)n-1ba 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)n-1t 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 × × 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 non-empty 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].

   1. 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].

   2. 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].

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

   4. 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.

4. 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 r-element 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 r-element 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 r-element 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.

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 × × 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

1. 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,

   1. xx xy and yy yx imply x = y,

   2. 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.

1. x a x b if and only if a x b x,

2. x x a x x b implies x a x b,

3. 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.

1. 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.

2. 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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