We prove that if a right modular groupoid U is cancellative or has a left
identity then a right modular generalised inflation of U is an inflation of U

Monzo, R. A. R.

English

arXiv

Abstract. We prove that if a right modular groupoid U is cancellative or has a left identity, then a right modular generalised inflation of U is an inflation of U

R A R Monzo

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Novi Sad J. Math.
Vol. 42, No. 2, 2012, 43-48
ON GENERALISED INFLATIONS OF RIGHT
MODULAR GROUPOIDS
R. A. R. Monzo1
Abstract. We prove that if a right modular groupoid U is cancellative
or has a left identity, then a right modular generalised inflation of U is
an inflation of U.
AMS Mathematics Subject Classification (2010): 18B40; 03C05
Key words and phrases: generalised inflation, right modular groupoid
1. Introduction
Since groupoids in the groupoid variety determined by the identity xy ·
z = zy · x were studied by Kazim and Nazeerudin in 1977 under the name
of left almost semigroups they have attracted a wide interest [6]. They have
appeared under the names of Abel Grassmann’s, left invertive and right modular
groupoids [1, 3–5, 7–9, 11–17]. The concept of a generalised inflation G of its
subsemigroup U was introduced in 2000 by Clarke and Monzo as an extension
of the concept of an inflation of a semigroup [2]. It had been known that a
semigroup S has a semilattice of groups as its square if and only if S is an
inflation of S2 and S2 is a semilattice of groups [18]. In [2], it was proved
that a semigroup S has a square that is a union of groups if and only if S is a
(symmetric) generalised inflation of S2 and S2 is a union of groups.
Wang and Wismath pointed out that the concept of a generalised inflation
can be extended to any algebra and therefore to groupoids; hence, the resultant
generalised inflation need not be a semigroup, even if U is a semigroup [19]. An
interesting result that links with those results on semigroups mentioned in the
above paragraph was obtained in [11], where it was proved that if a generalised
inflation G of its subgroupoid U is a right modular groupoid and U is a union
of groups, then G is an inflation of U.
In this paper we prove that a right modular generalised inflation G of a
cancellative groupoid U, or of a groupoid U with a left identity element, is an
inflation of U.
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 or x · y. We define x1 ≡ x and, by induction,
xn ≡ x · xn−1 for any n ∈ {2, 3,...}. For example, x4 = x ·
(
x · x2
)
. A groupoid
G will be called idempotent if x = x2 (x ∈ G). A groupoid G is called a right
modular groupoid if G satisfies the identity xy · z = zy · x.
1Flat 10 Albert Mansions, Crouch Hill, London N8 9RE, United Kingdom, e-mail: bob-
monzo@talktalk.net
44 R. A. R. Monzo
Definition 1. A groupoid [semigroup] G is an inflation of its subgroupoid
[subsemigroup] U if G =
∪
Gu (u ∈ U), where
(1) u ∈ Gu (u ∈ U),
(2) Gu ∩ Gv = ∅ if u ̸= v and
(3) x ∈ Gu and y ∈ Gv implies xy = uv.
Definition 2. A groupoid [semigroup] G is a generalised inflation of its sub-
groupoid [subsemigroup] U if G =
∪
Gu (u ∈ U), where
(1) u ∈ Gu (u ∈ U),
(2) Gu ∩ Gv = ∅ if u ̸= v,
(3) for every x ∈ Gu there exist αx and βx, right and left mappings re-
spectively on U, such that for every x ∈ Gu and y ∈ Gv, xy = vαx · βyu
and
(4) u ∈ U implies αu = κu = βu, where κu is a constant mapping that
sends every element to u.
Definition 3. A generalised inflation G of its subgroupoid [subsemigroup] U
is called a symmetric generalised inflation of its subgroupoid [subsemigroup] U
if uαx = βxu for every x ∈ G and u ∈ U. If a symmetric generalised inflation
G of U satisfies uαx = vαx for every x ∈ G and {u, v} ⊆ U then G is a constant
generalised inflation of U.
Result 2 below states that a constant generalised inflatation G of its sub-
groupoid U is an inflation of U. We omit the proof as it is straightforward.
Conversely, an inflation G of its subgroupoid U can be expressed as a con-
stant generalised inflation G of U. However, there are non-constant generalised
inflations G of U that are inflations of U. As stated in the introduction, if a
semigroup U is considered as a groupoid then it may have generalised inflations
that are not semigroups.
The following results will be used later in the proof of Theorems 1 and 3.
Result 1. ([6]) If G is a right modular groupoid and {a, b, c, d} ⊆ G then
ab · cd = ac · bd.
Result 2. If G is a constant generalised inflation of its subgroupoid U then G
is an inflation of U.
3. Generalised inflations of right modular groupoids
Although there may be many generalised inflations indeed of a given right
modular groupoid, if we require them to also be right modular groupoids, then
there are far fewer such generalised inflations. In certain cases, the only ones
are the inflations, as in the following example of the unique right modular,
idempotent groupoid of order 4. It is non-associative and is a quasigroup.
Non-associative right modular, idempotent groupoids of order ≤ 3 do not exist
[17].
On generalised inflations of right modular groupoids 45
Example 1. W = {1, 2, 3, 4} with multiplication table as follows.
W 1 2 3 3
1 1 3 4 2
2 4 2 1 3
3 2 4 3 1
4 3 1 2 4
Table 1: W = {1, 2, 3, 4}
Taking x /∈ W and H = W ∪ {x}, there are over 282,270,000,000,000,000
groupoids on the set H. If we require the multiplication in W to remain intact
then
1. there are 59 = 1, 953, 125 different groupoids that can be formed on the
set H, some of which may be isomorphic,
2. there are 49 = 262, 144 different groupoids H in which H2 ⊆ W and
3. there are less than 49 + 1 generalised inflations H of W.
Clearly, an inflation of a right modular groupoid is right modular. Which
of the generalised inflations H of W are right modular generalised inflations?
Since W is cancellative there are only 4, the inflations of W, as we now prove.
Theorem 1. If G is a right modular generalised inflation of its right cancella-
tive subgroupoid U then G is an inflation of U.
Proof. Let {u, v, w} ⊆ U and let x ∈ G − U. Then ux · v = vx · u and so
(uβxu) v = (vβxv)u = (uβxv) v. Therefore, uβxu = uβxv . Also, xu · v = vu ·x
and so (uαxu) v = (vu)βx (vu) = (vu)βxw = [(βxw)u] v. Therefore, uαx =
βxw, for every {u,w} ⊆ U. Thus, G is a constant generalised inflation of U
and, by Result 2, an inflation of U.
It is straightforward to show that a right modular groupoid is cancellative
if and only if it is right cancellative. Also, the following result has been proved
in [11].
Theorem 2. Let U be a right modular union of groups. Then any right modular
generalised inflation of U is an inflation of U. In fact, any right modular
groupoid G with G2 isomorphic to U is an inflation of G2.
Hence, any right modular generalised inflation of an idempotent, right mod-
ular groupoid U is an inflation of U. As noted above, a (semigroup) generalised
inflation S of its subsemigroup U, where U is a union of groups, need not be
an inflation of U.
Note that any semigroup S in which S2 has an identity element is an inflation
of S2. Consequently, any generalised inflation S of a subsemigroup U of the
semigroup S, in which U2 has an identity element, is an inflation of U2 = S2.
46 R. A. R. Monzo
This result holds because in the definition of a generalised inflation S of its
subsemigroup U, S is a semigroup. However, a generalised inflation G of a
semigroup U need not be a semigroup if we treat U as a groupoid, since the
definition of a generalised inflation G of its subgroupoid U does not require G
to be a semigroup.
Example 2. Consider the two-element chain C = {a, b}, where ab = ba = b,
a = a2 and b = b2. Let U = C1, G = U ∪ {x} (x /∈ U), 1αx = 1 = βx1, aαx =
b = bαx, βxa = a and βxb = b.
Then G is a generalised inflation of U, with x ∈ G1, but x2a = 1a = a ̸=
b = b2 = b · ba = bαx · [(aαx) a] = x ·xa and so G is not a semigroup. Note that
U is a right modular groupoid with an identity element. However, the groupoid
G is not right modular. The multiplication table for G = U∪{x} is as follows.
G a b 1 x
a a b a b
b b b b b
1 a b 1 1
x b b 1 1
Table 2: G = U ∪ {x}
This leaves us with the following natural question: If G is a right modular
generalised inflation of its subgroupoid U and U has an identity element, then
is G an inflation of U?
Theorem 3. If G is a right modular generalised inflation of its subgroupoid U
and U has a left identity element, 1 say, then G is an inflation of U.
Proof. Let {x, y} ⊆ G. Since G is a generalised inflation of U, G2 ⊆ U. Then,
using Result 1, xy = (1 · 1) · xy = 1x · 1y. Now, G =
∪
Gu (u ∈ U), where for
u ∈ U, Gu = {a ∈ G : 1a = u}, is a disjoint union satisfying u ∈ Gu (u ∈ U).
Also, for x ∈ Gu and y ∈ Gv ({u, v} ⊆ U) we have xy = uv. By definition
then, G is an inflation of U.
Theorem 4. If G is a right modular groupoid and x /∈ G then G ∪ {x} is a
right modular generalised inflation of G if and only if there exist right and left
mappings, αx and βx respectively, on G and an element c ∈ G such that for all
{a, b} ⊆ G,
(1) (aαxa) b = (ba)βx (ba),
(2) (aβxa) b = (bβxb) a and
(3) (cαx · βxc) · a = (aβxa) · βx (aβxa).
Proof. (⇒) Since G ∪ {x} is a right modular generalised inflation of G, the
mappings αx and βx exist. Let x ∈ Gc. The equations xa·b = ba·x, ax·b = bx·a
and x2 · a = ax · x are valid, since G∪ {x} is a right modular groupoid. By the
definition of a generalised inflation of G, (1), (2) and (3) follow.
On generalised inflations of right modular groupoids 47
(⇐) Define Ga = a (a ̸= c) and Gc = {c, x}. Then G∪{x} =
∪
Ga (a ∈ G),
with Ga∩Gb = ∅ if a ̸= b. Define dαc = c = βcd for any {c, d} ⊆ G. Then, since
(1), (2) and (3) are valid, we can use the α’s and β’s to define a generalised
inflation in which xa · b = ba · x, ax · b = bx · a and x2 · a = ax · x. Such a
generalised inflation satisfies the equation yz · w = wz · y and is therefore a
right modular generalised inflation of G.
It is straightforward to show that the four inflations W∪{x} of W, where W
is as in Example 1, are isomorphic. So of the 262,144 groupoids on H = W∪{x}
in which H2 ⊆ W, and therefore in which H may be a generalised inflation of W,
there is (to within isomorphism) only one right modular generalised inflation
of W. Clearly then, it is quite a powerful condition to require a generalised
inflation to be right modular.
Open Question
The author does not know if a semigroup G can be a generalised inflation
of its commutative subsemigroup U without being an inflation of U. Similarly,
can a right modular groupoid G be a generalised inflation of its (right modular)
subgroupoid U without being an inflation of U?
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Received by the editors March 7, 2011


ON GENERALISED INFLATIONS OF RIGHT MODULAR GROUPOIDS

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On Generalised Inflations of Right Modular Groupoids

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