Power groupoids and inclusion classes of Abel Grassmann groupoids

International Journal of Algebra, Vol. 5, 2011, no. 2, 91 - 105

Power groupoids and inclusion classes of Abel Grassmann groupoids R. A. R. Monzo Flat 10 Albert Mansions, Crouch Hill London N8 9RE, United Kingdom [email protected]

Abstract Within the groupoid variety AG = ⎡⎣( xy ) z = ( zy ) x ⎤⎦ of Abel Grassmann groupoids, we G determine the structure of all groupoids whose power groupoid is a band, a generalised inflation of a band or a union of groups. Such groupoids are semigroup chains, inflations of semigroup trees or

AG groupoids in the groupoid inclusion class ⎡⎣{( xy ) z , x ( yz )} ⊆ { x, y, z}⎤⎦ respectively. G We prove that G ∈ AG I ⎡⎣( xy )( zw ) ∈ { xy, xw, zy, zw}⎤⎦ if and only if its power groupoid G

P ( G ) is an AG inflation of an AG band if and only if P ( G ) is an AG generalised inflation of an AG band.

1. Introduction Groupoids in the (groupoid) variety AG = ⎡⎣( xy ) z = ( zy ) x ⎤⎦G have been called Abel Grassmann’s groupoids [10,11], left invertive groupoids [4] and LA- semigroups [5]. In this paper we explore how the structure of the power groupoids of groupoids in AG can determine their structure. Within the variety AG we show that the collection of all groupoids whose power groupoid is a band [generalised inflation of a band; a union of groups] determines an inclusion class of AG groupoids (cf. Theorems 1, 6 and 13). Similar work has been done with semigroups in [3], [6], [8], [9] and [12]. Section 2 contains preliminary definitions, notation and some known results used throughout the paper. In Section 3 we determine the structure of AG groupoids in which either the power groupoid or its square is a groupoid band. Such AG groupoids are semigroup chains or inflations of semigroup trees respectively. We also determine the structure of AG groupoids whose power

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groupoid is a union of groups. Results in this section are inspired by and build on the work of Redei [12], Pondelicek [9], Pelikan [8] and Monzo [6, 7].

2. Notation, definitions and preliminary results By a groupoid we shall mean a set G with a product ∗ : G × G → G and we shall denote ∗ ( x, y ) by xy . For x ∈ G we define x1 ≡ x and, by induction, x n ≡ x n −1 x for any n ∈ {2,3,...,} . For example, x 4 = ⎡⎣( xx ) x ⎤⎦ x . If U 2 ⊆ U ⊆ G then U is called a subgroupoid of G, which we denote by U ≤ G . Also G ≅ H will denote that G and H are isomorphic groupoids. A groupoid [semigroup] G is an inflation of its subgroupoid [subsemigroup] U if G = UGa ( a ∈U ) , where (1) a ∈ Ga for every a ∈U , (2) Ga I Gb = ∅ if a ≠ b and (3) for every x ∈ Ga and y ∈ Gb , xy = ab . In this case, for any x ∈ Ga ( a ∈U ) we define x$ ≡ a , and so xy = x$ $y for any { x, y} ⊆ G .

A groupoid [semigroup] G is a [symmetric] generalised inflation of its subgroupoid [subsemigroup] U if G = UGa ( a ∈U ) , where (1) a ∈ Ga for every

a ∈U , (2) Ga I Gb = ∅ if a ≠ b , (3) for every x ∈ Ga there exist α x and β x , right and left mappings respectively on U , such that xy = bα x β y a for every x ∈ Ga and y ∈ Gb and (4) a ∈U implies α a = κ a = β a , where κ a is a constant

mapping that sends every element to a [(5) α x = β x ( x ∈ U ) ]

An element a ∈ G is called idempotent [3-potent] if a = a 2 [ a 2 a = aa 2 ]. The groupoid G is called a groupoid band [3-potent] if every element of G is idempotent [3-potent]. Then EG ≡ { x ∈ G : xx = x} is the set of idempotents of G. If in addition a groupoid band is commutative then it will be called a groupoid semilattice. If {a, a '} ∈ G and a = ( aa ' ) a = a ( a ' a ) then a is called a regular element of G and a ' is called a partial inverse of a. If, in addition, a ' = ( a ' a ) a ' = a ' ( aa ') then

a ' is called an inverse of a. In this case a and a ' are called mutual inverses. The set of all partial inverses [inverses] of the element a is denoted by P ( a ) ⎡⎣V ( a ) ⎤⎦ .

Then Pc ( a ) ≡ {a ' ∈ P ( a ) : aa ' = a ' a} and Vc ( a ) ≡ {a ' ∈ V ( a ) : aa ' = a ' a} . If

Pc ( a ) ≠ ∅

then a is called a

∗

- regular element of G. Also

Pl ( a ) ≡ {a ' ∈ G : ( aa ') a = a} and Pr ( a ) ≡ {a ' ∈ G : a ( a ' a ) = a} are the sets of left and right partial inverses of a.

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The groupoid G is called a regular groupoid [ inverse groupoid ] if every element of G is regular [ has a unique inverse] . We will use REG [ INV ] to denote the collection of all regular [inverse] groupoids. If G satisfies the equation ( xy ) z = ( zy ) x then G is called an AG groupoid. We use AG to denote the collection of all AG groupoids, SEM to denote the collection of all semigroups andUG to denote the collection of all groupoids that are a union of (not necessarily disjoint) groups. The power groupoid of a groupoid G is defined as P ( G ) ≡ {AB : A , B ⊆ G}( A ≠ ∅ ≠ B ) , where AB = {ab : a ∈ A , b ∈ B} . A chain S is a semigroup in which, for any { x, y} ⊆ S , either x = xy = yx or y = xy = yx . The collection of all semigroup chains will be denoted byC . If

H i ⊆ Ki ( i ∈ {1, 2,..., n} ) is a finite collection of inclusions,

where H i , Ki ( i ∈ {1, 2,..., n} ) are meaningful groupoid {semigroup} words over

some alphabet, then [Η 1 ⊆ Κ 1 ;...; Η n ⊆ Κ n ]G { [Η 1 ⊆ Κ 1 ;...; Η n ⊆ Κ n ] } denotes

the collection of all groupoids {semigroups} that satisfy the inclusions H i ⊆ Ki ( i ∈1, 2,..., n ) and will be called an inclusion class of groupoids {semigroups}.

If I is any collection of groupoids then I N F

(I )

will denote the collection of

all inflations of groupoids in I . A groupoid G is called a groupoid semilattice if G ∈ ⎡⎣ x = x 2 ; xy = yx ⎤⎦ . A groupoid semilattice is called a groupoid tree if G

e = eg = ge and f = fg = gf implies ef = fe ∈ {e, f } . If a groupoid semilattice

[groupoid tree] G is also a semigroup then we call G a semilattice [tree].We denote the collection of all (semigroup) trees byT . The collection of all 3potent AG groupoids G ∈ ⎡⎣ x = x 2 x ⎤⎦ will be denoted as AG 2 . The following G

results are well known and will be used throughout this paper.

Result 1. [5] A commutative AG groupoid is a semigroup. Result 2. [5] An AG groupoid with a right identity element is commutative (and is therefore a semigroup). Result 3. [5] If G ∈ AG then for any { x, y, z, w} ⊆ G , ( xy )( zw) = ( xz )( yw ) . Result 4. If G ∈ AG then EG ≤ G . If EG ∈ ⎡⎣ xy ∈ { x, y}⎤⎦G then EG ∈C .

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Result 5. [11] Let G ∈ AG . Then G ∈ ⎡⎣ x = x 2 x = xx 2 ⎤⎦ if and only if G is an AG G band Y = EG of abelian groups of order 2. Result 6. [3,12] S ∈ ⎡⎣ xy ∈ { x, y}⎤⎦ iff S is a chain Y of semigroups

Sα ∈ [ xy = x ] U [ xy = y ] (α ∈Y ) , where x = xy = yx whenever x ∈ Sα , y ∈ S β and α = αβ = βα . Result 7. [1] S ∈ SEM is an inflation of a band iff S ∈ ⎡⎣ xy = x 2 y 2 ⎤⎦ . Result 8. [2] S ∈SEM is a generalised inflation of a band iff S ∈ [ xy = xyxy ] . Result 9. [13] If

S∈I N F

({ S } ) .

S ∈ SEM and S 2

is a semilattice of groups then

2

Result 10. G ∈ AG iff P (G ) ∈ AG . Result 11. [10] If G ∈ AG then the following statements are equivalent: (1) G is a generalised inflation of an AG band; (2) G is an inflation of an AG band; (3) G ∈ ⎡⎣ xy = x 2 y 2 ⎤⎦ and (4) G 2 ∈ ⎡⎣ x = x 2 ⎤⎦ I AG . G

G

Result 12. If G ∈ AG is a generalised inflation of U ≤ G then G 2 ≅ U 2 . Result 13. [7, Cor. 16] If G ∈ AG and G 2 ∈UG then G is an inflation of G 2 . Result 14. [7, Theorem. 12] If G ∈ AG then G ∈UG iff ∀x ∈ G , Pc ( x ) = 1 .

3. Power groupoids and inclusion classes of AG groupoids Clearly a semigroup S ∈ ⎡⎣ xy ∈ { x, y}⎤⎦ if and only if P ( S ) ∈ ⎡⎣ x = x 2 ⎤⎦ . Such semigroups are chains of left or right-zero semigroups (cf. [12], p.181, [3] and [6], 4.2). In [9] it was proved that a semigroup S ∈ ⎡⎣ xyzw ∈ { xy , xw, zy , zw}⎤⎦ if and only if ⎡⎣P ( S ) ⎤⎦ ∈ ⎣⎡ x = x 2 ⎦⎤ and the structure of such semigroups was completely described. With groupoids, P ( G ) ∈ ⎡⎣ x = x 2 ⎤⎦ if and only if G ∈ ⎡⎣ xy ∈ { x, y}⎤⎦ 2

G

and ⎡⎣P ( G ) ⎤⎦ ∈ ⎡⎣ x = x 2 ⎤⎦ if and only if G ∈ ⎡⎣( xy )( zw) ∈ { xy, xw, zy, zw}⎤⎦G . We G 2

G

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proceed to find all AG groupoids G in which P ( G ) is a groupoid band and those in which ⎡⎣P ( G ) ⎤⎦ is a groupoid band. 2

Theorem 1. ⎡⎣ xy ∈ { x, y}⎤⎦G I AG = C Proof: It is easy to see that any groupoid G contained in the intersection of the two groupoid inclusion classes is idempotent. Let { x, y, z} ⊆ G . Then

xy = x 2 y = ( yx ) x = yx and so G is commutative. By Result 1, G is a commutative semigroup. Since G ∈ ⎡⎣ xy ∈ { x, y}⎤⎦G and G is commutative, either x = xy = yx or y = xy = yx ; therefore G ∈C . Conversely, if G ∈C then clearly

G ∈ ⎡⎣ xy ∈ { x, y}⎤⎦G . Since G is commutative, G ∈ AG . ■

Theorem 2. ⎡⎣( xy )( zw ) ∈ { xy, xw, zy, zw}⎤⎦G I AG = = I N F (T

)

Proof. Let G ∈ ⎡⎣( xy )( zw ) ∈ { xy, xw, zy , zw}⎤⎦G I AG = . Let {e, f , g} ⊆ EgG . Then

fe = ( ef ) f = ( ef )( ff ) ∈ {ef , f } and so ef ∈ { fe, e} . Therefore either ef = fe or fe = f

and

ef = e .

But

ef = ( fe ) e

and

so

ef = fe

either

or

e = ef = ( fe ) e = fe = f . So EG is commutative and therefore is a semigroup. Furthermore, if e = eg = ge and f = fg = gf then

fe = ef = ( eg )( fg ) ∈ {eg , fg} = {e, f } and EG is a tree. Also, since for any

{a, b, c} ⊆ G , ( ab )( ab ) ∈ {ab} , EG = G 2 . So ab = ( ab )2 = a 2b 2 and 2 2 2 ( ab ) c = ( av ) c 2 = ⎡⎣ a 2b2 ⎤⎦ c 2 = ( a 2b2 ) c 2 = a 2 ( b2c 2 ) = a 2 ( b2c 2 ) = a ( bc ) and so G ∈ ⎡⎣ xy = x 2 y 2 ⎤⎦ . By Result 9, G ∈ I N F (T ) and we have shown that ⎡⎣( xy )( zw ) ∈ { xy , xw, zy, zw}⎤⎦G I AG = ⊆ I N F (T ) . Conversely, suppose that G ∈ INF (T ) . Then, since an inflation of a commutative semigroup is commutative, G ∈ AG . Then since G 2 ∈T , by Lemma 4.6 [6], G 2 ∈ ⎡⎣( xy )( zw ) ∈ { xy , xw, zy , zw}⎤⎦ . Clearly, since G is an inflation of G 2 , G ∈ ⎡⎣( xy )( zw ) ∈ { xy, xw, zy, zw}⎤⎦ ⊆ ⎡⎣( xy )( zw ) ∈ { xy, xw, zy, zw}⎤⎦ . So, G

I N F (T ) ⊆ ⎡⎣( xy )( zw ) ∈ { xy , xw, zy, zw}⎤⎦G I AG .■

Lemma 3. If G ∈ AG then P ( G ) is an inflation of an AG union of groups iff

P ( G ) is a generalised inflation of a union of groups iff ⎡⎣P ( G ) ⎤⎦ is a union of groups. 2

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Proof. Let G ∈ AG . By definition it is easy to see that if P ( G ) is an AG inflation

of a union of groups then P ( G ) is an AG generalised inflation of a union of groups, which by Result 12 implies ⎡⎣P ( G ) ⎤⎦ is a union of groups, which by Results 10 and 13 implies P ( G ) is an AG inflation of a union of groups. ■ 2

Lemma 4. G ∈ AG I ⎡⎣( xy )( zw ) ∈ { xz , xw, yz , yw}⎤⎦G iff P ( G ) is an AG inflation of an AG groupoid band. Proof. ( ⇒ ) Using Results 10 and 11, we must show that P ( G ) ∈ ⎡⎣ xy = x 2 y 2 ⎤⎦ . G

{a, a '} ⊆ A ⊆ G and {b, b '} ⊆ B ⊆ G . Then ( aa )( bb ) ∈ {ab} and so AB ⊆ A2 B 2 . Also, ( aa ')( bb ') ∈ {ab, ab ', a ' b, a ' b '} ⊆ AB and so A2 B 2 ⊆ AB ⊆ A2 B 2 . Hence, P ( G ) ∈ ⎡⎣ xy = x 2 y 2 ⎤⎦ . ( ⇐) By Result 11, G P ( G ) ∈ ⎡⎣ xy = x 2 y 2 ⎤⎦ . Let {a, b, c, d } ⊆ G and let A = {a, b} and B = {c, d } . G 2 2 Then ( ab )( cd ) ∈ A B = AB = {ac, ad , bc, bd } and so G ∈ ⎡⎣( xy )( zw ) ∈ { xz , xw, yz , yw}⎤⎦G . Since P ( G ) ∈ AG , by Result 10, Let

G ∈ AG . ■

Lemma 5. G ∈ AG I ⎡⎣( xy )( zw ) ∈ { xy, xw, zy, zw}⎤⎦G iff P ( G ) is an AG generalised inflation of an AG groupoid band. Proof. ( ⇒ ) Using Lemma 3 and Results 10 and 11, we need only show that ⎡⎣P ( G ) ⎤⎦ ∈ ⎡⎣ x = x 2 ⎤⎦ . As in the proof of Theorem 2, G ∈ ⎡⎣ xy = x 2 y 2 ⎤⎦ . G G 2 2 Therefore, for any A ⊆ G and B ⊆ G , AB ⊆ A B . Also, for any {a, a '} ⊆ A and 2

{b, b '} ⊆ B ,

( aa ')( bb ') = ( ab )( a ' b ') ∈ {ab ', ab ', a ' b, a ' b '} ⊆ AB . 2 2 A2 B 2 ⊆ AB ⊆ A2 B 2 and AB = A2 B 2 = ( AB ) and ⎡⎣P ( G ) ⎤⎦ ∈ ⎡⎣ x = x 2 ⎤⎦ . G

Hence,

( ⇐) By Lemma 3 and Results 10 and 11, ⎡⎣P ( G )⎤⎦ ∈ ⎡⎣ x = x 2 ⎤⎦G . Let {a, b, c, d } ⊆ G and let A = {a, c} and B = {b, d } . Then 2 ( ab )( cd ) ∈ ( AB ) = AB = {ab, ad , cb, cd } . So, G ∈ ⎡⎣( xy )( zw ) ∈ { xy, xw, zy, zw}⎤⎦G . Since P ( G ) ∈ AG , by Result 10, G ∈ AG . ■ 2

Theorem 6. The following statements are equivalent:

(1) G ∈ AG I ⎡⎣( xy )( zw ) ∈ { xy , xw, zy , zw}⎤⎦G

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(2) G ∈ AG I ⎡⎣( xy )( zw ) ∈ { xz , xw, yz , yw}⎤⎦G ; (3) G ∈ I N F (T ) ; (4) P ( G ) is an AG inflation of an AG band ; (5) P ( G ) is an AG generalised inflation of an AG band and (6) G ∈ AG and ⎡⎣P ( G ) ⎤⎦ is a groupoid band. 2

We proceed to determine the structure of groupoids in other inclusion classes of AG groupoids. Lemma 7. ⎣⎡ xy ∈ { x3 , y 3 }⎦⎤ = I N F

( ⎡⎣ xy ∈ { x, y}⎤⎦ ) .

Proof: Note that we are dealing with inclusion classes of semigroups, so that the products are associative. Suppose that S ∈ I N F ⎡⎣ xy ∈ { x, y}⎤⎦ . Then S is an

(

)

inflation of a band and so, by Result 7, S ∈ ⎡⎣ xy = x 2 y 2 ⎤⎦ . But S 2 ∈ ⎡⎣ xy ∈ { x, y}⎤⎦

and therefore xy = x 2 y 2 ∈ { x 2 , y 2 } . Hence, INF

( ⎡⎣ xy ∈ { x, y}⎤⎦ ) ⊆ ⎡⎣ xy ∈ {x , y }⎤⎦ . However, since S is an inflation of a 2

2

band, for every a ∈ S , a 2 = α 2 = α = α 3 = a 3 for some idempotent α . Therefore, I N F ⎡⎣ xy ∈ { x, y}⎤⎦ ⊆ ⎡⎣ xy ∈ { x3 , y 3 }⎤⎦ . Now suppose that S ∈ ⎡⎣ xy ∈ { x3 , y 3 }⎤⎦ .

(

)

Then a 2 = a3 ( a ∈ S ) and so ES = S 2 = {a ∈ S : a = a 2 } ∈ ⎡⎣ xy ∈ { x, y}⎤⎦ . For any

{a, b} ⊆ S ,

ab = a 2

if

then

a 2b2 = a ( ab ) b = a3b = a 2 ( ab ) = a 4 = a 2 = ab .

Similarly, if ab = b 2 then a 2b 2 = ab . Therefore, S ∈ ⎡⎣ xy = x 2 y 2 ⎤⎦ and so S is an inflation INF

of

the

band

S 2 ∈ ⎡⎣ xy ∈ { x, y}⎤⎦ .

We

have

shown

that

( ⎡⎣ xy ∈ { x, y}⎤⎦ ) ⊆ ⎡⎣ xy ∈ { x , y }⎤⎦ ⊆ I N F ( ⎡⎣ xy ∈ { x, y}⎤⎦ ) as required. ■ 3

3

Theorem 8. The following statements are equivalent:

(1) G ∈ I N F (C (2) G ∈ I N F

)

( ⎡⎣ xy ∈ { x, y}⎤⎦ I [ xy = yx]) = I N F ( ⎡⎣ xy ∈ { x, y}⎤⎦

(3) G ∈ ⎡⎣ xy ∈ { x 2 x, yy 2 }⎤⎦ I AG G

(4) G ∈ ⎡⎣ xy ∈ { x x, y y}⎤⎦ I AG G 3 3 (5) G ∈ ⎡⎣ xy ∈ { x , y }⎤⎦ I [ xy = yx ] . 2

2

G

I AG

)

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98

Proof. It follows easily from Result 4 that I N F (C ) = I N F ⎡⎣ xy ∈ { x, y}⎤⎦ I [ xy = yx ]

(

)

. It then follows easily from

Theorem 1 that (1) and (2) are equivalent. It is also easy to see that I N F (C ) ⊆ ⎡⎣ xy ∈ { x 2 x, yy 2 }⎤⎦ I ⎡⎣ xy ∈ { x 2 x, y 2 y}⎤⎦ I AG , so that (1) implies G G

(3) and (4). Assume (4). For a ∈ G , a 2 = a 2 a = ( a 2 a ) a = ( a 2 ) ∈ EG . For 2

{e, f } ⊆ EG , ef ∈ {e2 e, f 2 f } = {e, f } . Then fe = ( ef ) f = { eff ifif efef==fe EG ∈ ⎡⎣ xy ∈ { x, y}⎤⎦ I [ xy = yx ] = C . Since for any {a, b} ⊆ G ,

= ef and so

ab ∈ {a 2 a, b 2b} = {a 2 , b 2 } ⊆ EG and so, by Result 1, G 2 ∈C . Also xy = ( xy ) = x 2 y 2 . By Result 7, G ∈ I N F (C 2

)

and so (4) implies (1). Similarly, (3) implies (1). Hence, (1), (2), (3) and (4) are equivalent. Then, using Lemma 7, (1) through (5) are equivalent. ■ Theorem 9. ⎡⎣ xy = x 2 x ⎤⎦ I AG = I N F G

({0})

Proof. Let a ∈ G ∈ ⎡⎣ xy = x 2 x ⎤⎦ I AG . Then a 2 = a 2 a = ( a 2 ) ∈ EG . For any 2

G

{e, f } ⊆ G , ef = e2e = e and so e = ef = ( fe ) e = fe = f and EG = {0} . But then for any {a, b, c, d } ⊆ G , ab = a 2 a = a 2 = c 2 = c 2 c = cd . Thus, EG = G 2 = {0} . But since ab = a 2 = a 2b 2 = 0 , G is an inflation of G 2 = {0} , where x$ = 0 ( x ∈ G ) . So ⎡⎣ xy = x 2 x ⎤⎦ ⊆ I N F ({0} ) . It is straightforward to show that G I N F ({0} ) ⊆ ⎡⎣ xy = x 2 x ⎤⎦ . ■ G

Theorem 10. ⎡⎣( xy ) x ∈ { x, y}⎤⎦G I AG ⊆ AG 2 Proof. Let a ∈ G ∈ ⎡⎣( xy ) x ∈ { x, y}⎤⎦G I AG . Then a 2 a = a ,

a 2 = ( a 2 a ) a = ( a 2 ) and aa 2 = ( a 2 a ) a 2 ∈ {a 2 , a} . So either aa 2 = a or 2

aa 2 = a 2 = ( aa 2 ) a 2 = ( a 2 ) a = a 2 a = a . So a 2 a = a = aa 2 and G ∈ AG 2 . ■ 2

Lemma 11. Let G ∈ AG and P ( G ) ∈ REG . Then EG ≠ ∅ implies EG ∈C . Proof. Let A = {e, f } ⊆ EG . Then there exists B ⊆ G such that

( AB ) A = A ( BA) = A . Let b ∈ B . Then ( eb ) f ∈ ( AB ) A = A = {e, f } . CASE 1 : ( eb ) f = e . Note that e ( be ) ∈ A ( BA) = A = {e, f } and so

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e = ( eb ) f = ( fb ) e = ( fe )( be ) = ⎡⎣( fe )( be ) ⎤⎦ e = ⎡⎣ e ( be ) ⎤⎦ ( fe ) ∈ {e ( fe ) , f ( fe )} .

So, e = ( eb ) f = ( ef )( bf ) and ef = ⎡⎣( ef )( bf ) ⎤⎦ f = ⎡⎣ f ( bf ) ⎤⎦ ( ef ) ∈ {e ( ef ) , f ( ef )} since f ( bf ) ∈ A ( BA) = A = {e, f } . CASE 1.1 : ef = e ( ef ) . Note that e = ( eb ) f = ⎡⎣( eb ) f ⎤⎦ e = ( ef )( eb ) = e ( fb ) .

{

}

Therefore, ef = e ( ef ) = e ⎡⎣( eb ) f ⎤⎦ f = e ⎡⎣ f ( eb ) ⎤⎦ = e ⎡⎣( fe )( fb ) ⎤⎦ =

= ⎡⎣e ( fe ) ⎤⎦ ⎡⎣e ( fb ) ⎤⎦ = ⎡⎣e ( fe ) ⎤⎦ e . But, since e ∈ {e ( fe ) , f ( fe )} , we can assume that

e = f ( fe ) or else ef = e . Now eb = ⎡⎣ f ( fe ) ⎤⎦ b = ⎡⎣b ( fe ) ⎤⎦ f and so

{

} {

}

f ( eb ) = f ⎡⎣b ( fe ) ⎤⎦ f = f ⎡⎣b ( fe ) ⎤⎦ f =

{( fb ) ⎡⎣ f ( fe )⎤⎦} f = ⎡⎣( fb ) e⎤⎦ f = ef .

Then, e = ee = ⎡⎣ f ( fe ) ⎤⎦ ⎡⎣ e ( fb ) ⎤⎦ = ( fe ) ⎡⎣( fe )( fb ) ⎤⎦ = ( fe ) ⎡⎣ f ( eb ) ⎤⎦ = ( fe )( ef ) , so fe = f ⎡⎣( fe )( ef ) ⎤⎦ = ⎡⎣ f ( fe ) ⎤⎦ ⎡⎣ f ( ef ) ⎤⎦ = e ⎡⎣ f ( ef ) ⎤⎦ = e ⎡⎣( fe ) f ⎤⎦ = ⎡⎣ e ( fe ) ⎤⎦ ( ef ) = = ⎡⎣e ( fe ) ⎤⎦ ⎡⎣e ( ef ) ⎤⎦ = e ⎡⎣( fe )( ef ) ⎤⎦ = e. Thus, fe = e = ee = ( fe ) e = ef . CASE 1.2 : ef = f ( ef ) = ( fe ) f . Then fe = ( ef ) f = ⎡⎣( fe ) f ⎤⎦ f = f ( fe ) . But since e ∈ {e ( fe ) , f ( fe )} we can assume that e = e ( fe ) = ( ef ) e or else

ef = ( fe ) e = e . Then, e = ( eb ) f = ( fb ) e = ( fe )( be ) and

so fe = f ⎡⎣( fe )( be ) ⎤⎦ = ⎡⎣ f ( fe ) ⎤⎦ ⎡⎣ f ( be ) ⎤⎦ = = ( fe ) ⎡⎣ f ( be ) ⎤⎦ ∈ {( fe ) f , ( fe ) e} = {ef } . So e = ( ef ) e = ( fe ) e = ef . CASE 2 : ( eb ) f = f . Then f = ( eb ) f = ( fb ) e = ( fe )( be ) = ⎡⎣( fe )( be ) ⎤⎦ f = ⎡⎣ f ( be ) ⎤⎦ ( fe ) ∈ {e ( fe ) , f ( fe )} .

So fe = ⎡⎣( eb ) f ⎤⎦ e = ⎡⎣( eb ) e ⎤⎦ ( fe ) = ( fe ) = ( fe ) ⎡⎣( eb ) e ⎤⎦ ∈ {( fe ) e, ( fe ) f } . 2

CASE 2.1 : fe = ( fe ) f = f ( ef ) . Note that fe = ( fe ) f = ⎡⎣( fe ) f ⎤⎦ f = f ( fe ) . If f = f ( fe ) then fe = f ( fe ) = f , which implies that ef = ( fe ) e = fe = f . So we

can assume that f = e ( fe ) = ( ef ) e . Then f = ff = ⎡⎣ e ( fe ) ⎤⎦ f = ⎡⎣ f ( fe ) ⎤⎦ e = ( fe ) e = ef .

CASE 2.2 : fe = ( fe ) e = ef . Then, if f = e ( fe ) = ( ef ) e , ef = fe = ⎡⎣( ef ) e ⎤⎦ e = e ( ef ) = e ( fe ) = f and so we can assume that f = f ( fe ) .

Then, f = f ( fe ) = ⎡⎣ f ( fe ) ⎤⎦ ( fe ) = ( fe ) f = ( ef ) f = fe , which implies that

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ef = ( ef ) = ( fe ) = f 2 = f . So we have shown that ef ∈ {e, f } and, by Result 4, 2

2

if EG ≠ ∅ then EG ∈C . ■ Theorem 12. G ∈ AG and P ( G ) ∈UG

implies G ∈C or G = EG U {q} where

EG ∈C , q ∉ EG , qe = eq = e for every e ∈ EG − {q 2 } and q 2 x = xq 2 = x for every

x ∈G . Proof. First we will prove that every element of G has a partial inverse with which it commutes, which by Result 14 implies that G ∈UG .

Let a ∈ G , with A = {a} . Then since P ( G ) ∈UG there exists B ⊆ G such that

( AB) A = A ( BA ) = A Then ( ab ) a = a ( ba ) = a ( ba ) a = a and ab = ( ba )2 .

and

AB = BA = ( AB ) = ( BA ) . 2

Let b ∈ B .

2

and ( ba ) a ∈ ( BA ) A = ( AB ) A = A .

So

Also, a 2 a = a 2 ⎡⎣( ba ) a ⎤⎦ = ⎡⎣ a ( ba ) ⎤⎦ a 2 = aa 2 . So for any

x ∈ G we can show similarly that x 2 x = xx 2 . Then 2 2 ab = ⎡⎣( ab ) a ⎤⎦ b = ( ba )( ab ) = ( ba )( ba ) = ( ba ) ( ba ) = ( ab )( ba )

and

so

ab = ⎡⎣( ab )( ba ) ⎤⎦ ( ba ) = ( ba ) ( ab ) = ( ab ) . Similarly, there exists b ' ∈ G such that 2

2

b = ( bb ') b and bb ' ∈ EG . Then, ba = ( ab )( bb ') ∈ EG and so ab = ( ba ) = ba . 2

Hence, G ∈UG and by Lemma 11, ∅ ≠ EG ∈C . We now show that for any a ∈ G , a 2 ∈ EG . Let A = {a, a 2 } and let B ∈P ( G ) be such that B ∈ Vc ( A ) . If b ∈ B then ( a 2b ) a = ⎡⎣( ba ) a ⎤⎦ a ∈ ( AB ) A = A = {a, a 2 } . But

( ba ) a ∈ ( BA ) A = ( AB ) A = A = {a, a 2 } and so ( a 2b ) a ∈ {a 2 , a 2 a} . If

then a 2 a = a 2 ( a 2b ) ∈ {a, a 2 } , which implies that a 2 a 2 = a 2 ( a 2b ) . But a 2b = ( ba ) a ∈ {a, a 2 } and so a 2 = a 2

(a b) a = a ∈ E . If ( a b ) a = a then ( a b ) ∈ {a a, ( a ) } , which 2

2

2

G

2

2

2 2

implies that a 2 ∈ EG . We now prove that for any { x, e} ⊆ G satisfying x ≠ x 2 ≠ e ∈ EG , e = ex = xe and e = x 2 e = ex 2 . Let A = { x, e} and let B ∈ Vc ( A ) , with b ∈ B . Then ( eb ) x ∈ { x, e} .

Suppose that ( eb ) x = x . Now eb = ( be ) e ∈ { x, e} . But this implies that eb = e , or else x = ( eb ) x = x 2 , a contradiction. Hence, x = ( eb ) x = ex . Then

xb = ( ex ) b = ( bx ) e ∈ { x, e} , which implies xb = x , or else

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x = ( eb ) x = ( xb ) e = e = e2 = x 2 . Then x 2 = ( xb ) x ∈ { x, e} , a contradiction. So we can assume that ( eb ) x = e . Now eb = ( be ) e ∈ { x, e} , which implies that eb = e or else e = ( eb ) x = x 2 , a contradiction. So e = ( eb ) x = ex . Now xe = x ( eb ) ∈ { x, e} , which implies xe = e or else e = ex = ( xe ) e = xe = x , a contradiction. So we have shown that e = ex = xe. (As a consequence, e = ex 2 = x 2 e .) We now prove that G − EG ≤ 1. Let { x, y} ⊆ G − EG , with x ≠ y. We have already

proved that { x 2 , y 2 } ⊆ EG and we know that x ≠ x 2 and y ≠ y 2 . In the paragraph above

we

proved

that

if x 2 ≠ y 2 ,

Therefore, y 2 = y 2 x = ( y 2 x ) x = x 2 y 2

then y 2 = y 2 x = xy 2 and x 2 = x 2 y = yx 2 .

and similarly x 2 = y 2 x 2 . But this is a

contradiction , since EG ∈C implies that x 2 y 2 = y 2 x 2 . So x 2 = y 2 , which means

that any two non-idempotent elements of G are in the set Ga = { g ∈ G : g 2 = a} ,

where a ∈ EG and a is maximal in EG . We have already shown that G ∈UG and since e = xe = ex for every e ∈ EG (when e ≠ x 2 ) and x ≠ x 2 , x = xx 2 = x 2 x for every x ∈ Ga − EG . But then it is easy to see that Ga is an abelian group of order two.

Thus, { x, y} ∈ Ga ,

Take A = {a, x, y}

with a = 1Ga .

and

let B ⊆ G satisfy B ∈ Vc ( A ) . Then for b ∈ B , ( ab ) a ∈ {a, x, y} . This implies that b 2 = a and hence, b ∈ Ga . If ( ab ) a = a then b = a and therefore

( xb ) y = xy ∈ ( AB) A = {a, x, y} . If ( ab ) a = x then b = x and xy = ( ax ) y = ( ab ) y ∈ {a, x, y} . If ( ab ) a = y then b = y and xy = ( xy ) a = ( xb ) a ∈ {a, x, y} . We have therefore shown that xy ∈ {a, x, y} .

But

xy = a implies x = y , xy = x implies y = a and xy = y implies x = a . Hence we

have a contradiction to the hypothesis that { x, y} ⊆ G − EG and x ≠ y and therefore

G − EG ≤ 1. ■ Theorem 13. G ∈ AG and P ( G ) ∈UG if and only if

G ∈ ⎡⎣{( xy ) z, x ( yz )} ⊆ { x, y, z}⎤⎦ I AG G

Proof. ( ⇒ ) It follows easily from Theorem 12 that

G ∈ ⎡⎣{( xy ) z, x ( yz )} ⊆ { x, y, z}⎤⎦ I AG .

( ⇐)

G

Let {e, f } ⊆ EG . Then ef = ( fe ) e ∈ {e, f } , so by Result 4, EG ∈C . Note that

for any z ∈ G , z = z 2 z = zz 2 and z 2 = ( z 2 z ) z = ( z 2 ) ∈ EG . Assume that 2

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{ x, y} ⊆ G − EG and x ≠ y . Then ( xy ) y = y 2 x = ( y 2 y 2 ) x ∈ { x, y} I { x, y 2 } . Since x ≠ y and y ≠ y 2 , ( xy ) y = x . Similarly, ( yx ) x = y . But then y = ( yx ) x = {⎡⎣( yx ) x ⎤⎦ x} x = ⎡⎣ x 2 ( yx ) ⎤⎦ x = ⎡⎣( xy ) x 2 ⎤⎦ x ∈ { x 2 , yx, x} I { xy, x 2 , x} .

But y ∉ { x, x 2 } and so y = xy = yx . By symmetry, x = yx = xy and so x = y. We have shown that either G = EG ∈C or G = EG U {q} , where q ∉ EG . Clearly, qq 2 = q 2 q = q .

Then q 2 e ∈ EG and q 2 e = ( q 2 q 2 ) e ∈ {e, q} ,

Let e ∈ EG .

which

implies that q 2 e = e . Similarly, eq 2 = e . So we have shown that q 2 x = xq 2 = x for all x ∈ G . If e ≠ q 2 then qe = q ( ee ) ∈ {q, e} . If qe = q then q = qe = ( qe ) e = eq and then q 2 = ( qe ) q ∈ {e, q} , a contradiction. Hence, qe = e . Similarly, eq = e . We have shown that G is commutative and so G ∈ SEM . For any A ∈P ( G ) , clearly

A ( AA ) = ( AA ) A ⊆ A . Then A ⊆ ( AA ) A = A ( AA ) if q ∉ A . If q ∈ A then, since q = q 2 q = qq 2 , it is still the case that A ⊆ ( AA ) A = A ( AA ) . Then by Result 5 and 10, P ( G ) ∈UG . ■ Note that in the proof of Theorem 13 we have also proved the following. Corollary 14. If G ∈ AG and G has the form described in Theorem 12 then P ( G ) ∈UG I ⎡⎣ x = x 2 x = xx 2 ⎤⎦ . G

Corollary 15. G ∈ ⎡⎣{( xy ) z, x ( yz )} ⊆ { x, y, z}⎤⎦ I AG implies that G has the form G

described in Theorem 12. Lemma 16. Let G ∈ AG . Let Sn be the following statement : Any product P of 2n+1 factors ( a1 , a2 ,..., a2 n +1 say) in G is equal to a product P ' of the same factors

in which P ' contains a product of the form a ( bc ) or

( ab ) c ,

where

{a, b, c} ⊆ {a1 , a2 ,..., a2n+1} . Then Sn is true for every positive integer n . Proof. (By induction on n.) Clearly, S1 is true. Let n ≥ 2 and assume that St is true for all t ≤ n − 1 . Let P be any product, in G, of the 2 n +1 factors a1 , a2 ,..., a2 n +1 listed in order of their appearance in P. CASE 1:[ P begins in ( a1a2 ) P' ]. If P' has an odd number of factors and, by the

induction hypothesis, P' (and, therefore,P) contains a product of the required form. If P ' has an even number of factors then P begins in

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( a1a2 ) P' = ( P 'a2 ) a1 , and by the induction hypothesis P 'a2 contains a product of the required form, so that P is equal to a product containing a product of the required form. CASE 2:[P begins in a1P' = a1 ( AB ) ] We can assume that A and B both have an

even number of factors. We can assume that A has at least four factors, or else ( AB) = ( a2 a3 ) B = ( Ba3 ) a2 , which by the induction hypothesis contains a factor of the required form. Then A = A1B1 and again we can assume that A1 has an even number of factors, but not two factors, and B1 also has an even number of factors. Continuing in this manner we eventually obtain a factor A s Bs , where A s = ak ak +1 and Bs has an even number of factors. Then A s Bs = ( Bs ak +1 ) ak and Bs ak +1 has a

factor of the required form . Hence P is equal to a product of the a j ' s ( j ∈ {1, 2,..., 2n + 1} ) and this product has a factor of the required form. ■ Lemma 17. Let G ∈ AG and n ∈ N = {1, 2,...} . Let A n be the following statement :

Any product of n factors equals one of the factors. Then for any n ∈ N , A 2 is valid if and only if A 2n is valid and A3 is valid if and only if A 2n +1 is valid. Proof. [ A 2n ⇒ A 2 ] Let {a, b} ⊆ G . Then by A2 n , a 2 n = a . So, again by using

A2 n , a 2 a = a 2 a 2 n ∈ {a, a 2 } . This implies that a 2 = ( a 2 ) . Then a 2 = ( a 2 ) , which 2

n

by A2 n equals a. That is, a = a 2 for any a ∈ G . Using this fact and applying A2 n again gives ab = ( ab ) ∈ {a, b} and so A2 is valid. n

[ A 2 ⇒ A n , for any n ∈ N , n ≥ 3 ] Let {a, b, c} ⊆ G and assume that A 2 is valid .

Then ( ab ) c ∈ {ab, c} ⊆ {a, b, c} and a ( bc ) ∈ {a, bc} ⊆ {a, b, c} .

So A 2 implies A3 .

Assume that A t is valid for all t ∈ N , 3 ≤ t ≤ n − 1 . Let {a1 , a2 ,...aq , aq +1 ,..., an } be the factors of an arbitrary product T = AB , where T has n factors, a1 , a2 ,..., aq are the factors of A and aq +1 , aq + 2 ,..., an are the factors of B . Since, by the induction

hypothesis, A 2 , A q and A n-q are valid, T ∈ {A,B} ⊆ {a1 , a2 ,..., an } and so A n is valid. Note that we have proved that for any n ∈ N , A 2 n implies A 2 implies A 2n . [ A 2 n +1 ⇒ A 3 , for any n ∈ N ] Let a ∈ G . Then a 2 a = a 2 a 2 n +1 ∈ {a, a 2 } which

implies that a 2 ∈ EG . Then since a 2 a = ( a 2 ) a = a = a ( a 2 ) = aa 2 , by Result 5 n

n

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G is a union of groups of order 2. Let {u, v, w} ⊆ G . Then

( uv ) w = ( uv ) ( ww2 ) = ( uv )

{w ⎡⎣⎢( w ) ⎤⎦⎥}∈{u, v, w} .and 2 n −1

similarly, u ( vw ) ∈ {u, v, w} . Hence, A3 is valid.

[ A3 ⇒ A 2n+1 , for any n ∈ N .] Using Lemma 16, the proof of this by induction is straightforward. Theorem 18. If G ∈ AG then the following statements are equivalent.

(1) P ( G ) ∈UG ; (2) G ∈C or G = EG U {q} where EG ∈C , q ∉ EG , qe = eq = e ( q 2 ≠ e ∈ EG ) and q 2 x = x = xq 2 ( x ∈ G ) ;

(3) G ∈ ⎡⎣{ x ( yz ) , ( xy ) z} ⊆ { x, y, z}⎤⎦ I AG G 2 2 (4) P ( G ) ∈ ⎡⎣ x = x x = xx ⎤⎦ I AG ; G

(5) G ∈ AG and satisfies A 2 n +1 for some n ∈ N and (6) G ∈ AG and satisfies A 2 n +1 for every n ∈ N . Proof. [ 1 ⇔ 3 ] This is Theorem 13. [ 1 ⇔ 2 ] This follows from Theorem 12 and Corollary 14. [ 1 ⇔ 5 ⇔ 6 ] This follows from Lemmas 16 and 17. [ 2 ⇒ 4 ] This follows from Corollary 14 and Result 10. [ 4 ⇒ 1 ] This follows from Result 5. ■

Open Questions 1. Does there exist a symmetric generalised inflation S of a commutative semigroup U such that S is not an inflation of U ? 2. Does there exist a symmetric generalised inflation G ∈ AG of a groupoid U ∈ AG such that G is not an inflation of U ? 3. If G ∈ AG does P ( G ) ∈ REG imply P ( G ) ∈UG ?

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References 1. Clarke, G.T.: Semigroup varieties of inflations of unions of groups. Semigroup Forum 23, 311-319 (1981) 2. Clarke, G.T., Monzo, R.A.R.: A generalisation of the concept of an inflation of a semigroup. Semigroup Forum 60, 172-186 (2000) 3. Giraldes, E.: Howie, J.M.: Semigroups of high rank. Proc. Edinb. Math. Soc. 28, 13-34 (1985) 4. Holgate, P.: Groupoids satisfying a simple invertive law. Math. Student 61, 101-106 (1992) 5. Kazim, M.A., Naseeruddin, M.: On almost semigroups, The Aligarh Bull. Math., 2, 1-7 (1972) 6. Monzo, R.A.R.: Further results in the theory of generalised inflations of semigroups. Semigroup Forum 76, 540-560 (2008) 7. Monzo, R.A.R.: On the Structure of Abel Grassmann Unions of Groups International Journal of Algebra, 4(no.26), 1261-1275 (2010) 8. Pelikan, J.: On semigroups in which products are equal to one of the factors. Period. Math. Hung., 4, 103-106 (1973) 9. Pondelicek, B.: On semigroups having regular globals. Colloq. Mathem. Soc. Janos. Bolyai (1976) 10. Protic, P.V., Stevanovic, N.: Inflations of the AG groupoids, Novi Sad J. Math., 29(1), 19-26 (1999) 11. Protic, P.V., Stevanovic, N.: Composition of Abel Grassman’s 3-bands. Novi Sad J. Math., 34(2), 175-182 (2004) 12. Redei, L.: Algebra I. Pergamon, Elmsford (1967) 13. Tamura, T.: Semigroups satisfying identity xy = f ( x, y ) . Pac. J. Math. 31, 513-521 (1962) Received: August, 2010