EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 3, 2018, 876-881
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Introducing Partial Transformation UP-Algebras∗

Aiyared Iampan1,∗, Phakawat Mosrijai1, Akarachai Satirad1

1 Department of Mathematics, School of Science, University of Phayao, Phayao 56000,
Thailand

Abstract. The main aim of this paper is to introduce the notion of a partial transformation UP-
algebra P (X) induced by a UP-algebra X and prove that the set of all full transformations T (X)
is a UP-ideal of P (X).

2010 Mathematics Subject Classifications: 03G25
Key Words and Phrases: UP-algebra, partial transformation, full transformation.

1. Introduction and Preliminaries

Iampan [2] introduced a new algebraic structure, called a UP-algebra, which is a gener-
alization of a KU-algebra. Many researchers have studied on UP-algebras such as [4, 6, 7].
Let X be a universal set and let Ω ∈ P(X). Denote PΩ(X) = {A ∈ P(X) | Ω ⊆ A} and
PΩ(X) = {A ∈ P(X) | A ⊆ Ω}. Define a binary operation · on PΩ(X) by putting

A ·B = B ∩ (A′ ∪ Ω) for all A,B ∈ PΩ(X)

and a binary operation ∗ on PΩ(X) by putting

A ∗B = B ∪ (A′ ∩ Ω) for all A,B ∈ PΩ(X).

Satirad et al. [5] proved that (PΩ(X), ·,Ω) and (PΩ(X), ∗,Ω) are UP-algebras. In partic-
ular, (P(X), ·, ∅) and (P(X), ∗, X) are UP-algebras.

In this paper, we introduce the notion of a partial transformation UP-algebra P (X)
induced by a UP-algebra X and prove that the set of all full transformations T (X) is a
UP-ideal of P (X).

Now we will recall the definition of a UP-algebra from [2].

An algebra X = (X, ·, 0) of type (2, 0) is called a UP-algebra where X is a nonempty
set, · is a binary operation on X, and 0 is a fixed element of X (i.e., a nullary operation)
if it satisfies the following axioms: for any x, y, z ∈ X,

∗This work was financially supported by the University of Phayao.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3296

Email addresses: aiyared.ia@up.ac.th (A. Iampan),
phakawat.mo@gmail.com (P. Mosrijai), akarachai.sa@gmail.com (A. Satirad)

http://www.ejpam.com 876 c© 2018 EJPAM All rights reserved.



A. Iampan, P. Mosrijai, A. Satirad / Eur. J. Pure Appl. Math, 11 (3) (2018), 876-881 877

(UP-1) (y · z) · ((x · y) · (x · z)) = 0,

(UP-2) 0 · x = x,

(UP-3) x · 0 = 0, and

(UP-4) x · y = 0 and y · x = 0 imply x = y.

In a UP-algebra X = (X, ·, 0), the following assertions are valid (see [2, 3]).

(∀x ∈ X)(x · x = 0), (1.1)

(∀x, y, z ∈ X)(x · y = 0, y · z = 0⇒ x · z = 0), (1.2)

(∀x, y, z ∈ X)(x · y = 0⇒ (z · x) · (z · y) = 0), (1.3)

(∀x, y, z ∈ X)(x · y = 0⇒ (y · z) · (x · z) = 0), (1.4)

(∀x, y ∈ X)(x · (y · x) = 0), (1.5)

(∀x, y ∈ X)((y · x) · x = 0⇔ x = y · x), (1.6)

(∀x, y ∈ X)(x · (y · y) = 0), (1.7)

(∀a, x, y, z ∈ X)((x · (y · z)) · (x · ((a · y) · (a · z))) = 0), (1.8)

(∀a, x, y, z ∈ X)((((a · x) · (a · y)) · z) · ((x · y) · z) = 0), (1.9)

(∀x, y, z ∈ X)(((x · y) · z) · (y · z) = 0), (1.10)

(∀x, y, z ∈ X)(x · y = 0⇒ x · (z · y) = 0), (1.11)

(∀x, y, z ∈ X)(((x · y) · z) · (x · (y · z)) = 0), and (1.12)

(∀a, x, y, z ∈ X)(((x · y) · z) · (y · (a · z)) = 0). (1.13)

From now on, X will always denote a UP-algebra (X, ·, 0).

Definition 1. [2] A subset S of X is called a UP-subalgebra of X if the constant 0 of X
is in S, and (S, ·, 0) itself forms a UP-algebra.

Iampan [2] proved the useful criteria that a nonempty subset S of a UP-algebra X is
a UP-subalgebra of X if and only if S is closed under the · multiplication on X.

Definition 2. [2, 8] A subset S of X is called

(1) a UP-filter of X if it satisfies the following properties:

(i) the constant 0 of X is in S, and

(ii) for any x, y ∈ X,x · y ∈ S and x ∈ S imply y ∈ S.

(2) a UP-ideal of X if it satisfies the following properties:

(i) the constant 0 of X is in S, and

(ii) for any x, y, z ∈ X,x · (y · z) ∈ S and y ∈ S imply x · z ∈ S.

(3) a strongly UP-ideal of X if it satisfies the following properties:



A. Iampan, P. Mosrijai, A. Satirad / Eur. J. Pure Appl. Math, 11 (3) (2018), 876-881 878

(i) the constant 0 of X is in S, and

(ii) for any x, y, z ∈ X, (z · y) · (z · x) ∈ S and y ∈ S imply x ∈ S.

Guntasow et al. [1] proved the generalization that the notion of UP-subalgebras is a
generalization of UP-filters, the notion of UP-filters is a generalization of UP-ideals, and
the notion of UP-ideals is a generalization of strongly UP-ideals. Moreover, they also
proved that a UP-algebra X is the only one strongly UP-ideal of itself.

2. Main Results

We denote

B(X) the set of all binary relations on X,

P (X) the set of all partial transformations on X,

T (X) the set of all full transformations on X.

Then T (X) ⊆ P (X) ⊆ B(X). If α ∈ B(X) and x ∈ X, then xα = {y ∈ X | (x, y) ∈ α}.
Thus xα is the set of all elements that are α-related to x. Define a function O from X to
X by O(x) = 0 for all x ∈ X, that is, O ∈ T (X). Define a binary operation • on B(X)
by: for all α, β ∈ B(X),

(x, y) ∈ α • β ⇔
{
x ∈ domα ∩ domβ and y = yxα · yxβ for yxα ∈ xα and yxβ ∈ xβ, or
x /∈ domα and y = 0.

We can redefine a binary operation • on P (X) by: for all α, β ∈ P (X),

(α • β)(x) =

{
α(x) · β(x) if x ∈ domα ∩ domβ,
0 if x /∈ domα.

We see that

• for all α, β ∈ B(X),
dom (α • β) = (domα− domβ)

′
, (2.1)

• the empty function ∅ ∈ P (X) and for all α ∈ P (X),

∅ • α = O and α • ∅ = O|(domα)′ . (2.2)

Theorem 1. B(X) = (B(X), •, O) is an algebra of type (2, 0) satisfying (UP-2) and
(UP-3).

Proof. Let α ∈ B(X). Then

(x, y) ∈ O • α⇔ x ∈ X ∩ domα and y = rxO · yxα for some yxα ∈ xα (domO = X)

⇔ x ∈ domα and y = O(x) · yxα for some yxα ∈ xα



A. Iampan, P. Mosrijai, A. Satirad / Eur. J. Pure Appl. Math, 11 (3) (2018), 876-881 879

⇔ x ∈ domα and y = 0 · yxα for some yxα ∈ xα
⇔ x ∈ domα and y = yxα for some yxα ∈ xα ((UP-2))

⇔ (x, y) ∈ α.

Hence, O • α = α, so (UP-2) is holding.

Let α ∈ B(X) and x ∈ X. Then
Case 1: x /∈ domα. Then (x, 0) ∈ (α •O)⇔ (x, 0) ∈ O.
Case 2: x ∈ domα. Then

(x, y) ∈ α •O ⇔ x ∈ domα ∩X and y = rxα · yxO for some yxO ∈ xO (domO = X)

⇔ x ∈ domα and y = rxα ·O(x)

⇔ x ∈ domα and y = rxα · 0
⇔ x ∈ domα and y = 0 ((UP-3))

⇔ (x, y) ∈ O.

Hence, α •O = O, so (UP-3) is holding.

Therefore, B(X) = (B(X), •, O) is an algebra of type (2,0) satisfying (UP-2) and
(UP-3).

Theorem 2. P (X) = (P (X), •, O) is a UP-algebra and we shall call it the partial trans-
formation UP-algebra induced by a UP-algebra X.

Proof. Let α, β, γ ∈ P (X) and let x ∈ X.
Case 1: x /∈ domα. Then (α•β)(x) = 0 = (α•γ)(x), so x ∈ dom (α•β)∩dom (α•γ).

Thus

((α • β) • (α • γ))(x) = (α • β)(x) · (α • γ)(x)

= 0 · 0
= 0, ((UP-2))

so x ∈ dom ((α • β) • (α • γ)).
Case 1.1: x /∈ dom (β • γ). Then ((β • γ) • ((α • β) • (α • γ)))(x) = 0 = O(x).
Case 1.2: x ∈ dom (β • γ). Then x ∈ dom (β • γ) ∩ dom ((α • β) • (α • γ)). Thus

((β • γ) • ((α • β) • (α • γ)))(x) = (β • γ)(x) · ((α • β) • (α • γ))(x)

= (β • γ)(x) · 0
= 0 ((UP-3))

= O(x).

Case 2: x ∈ domα.



A. Iampan, P. Mosrijai, A. Satirad / Eur. J. Pure Appl. Math, 11 (3) (2018), 876-881 880

Case 2.1: x /∈ domβ. Then x ∈ domα− domβ, so (β • γ)(x) = 0 and (α • β)(x) is
not defined. Thus x /∈ dom (α • β), so ((α • β) • (α • γ))(x) = 0. Thus x ∈ dom (β • γ) ∩
((α • β) • (α • γ)), so

((β • γ) • ((α • β) • (α • γ)))(x) = (β • γ)(x) · ((α • β) • (α • γ))(x)

= 0 · 0
= 0 ((UP-2))

= O(x).

Case 2.2: x ∈ domβ. If x /∈ dom γ, then x ∈ domβ − dom γ. Thus (β • γ)(x) is not
defined, so x /∈ dom (β •γ). Thus ((β •γ)• ((α•β)• (α•γ)))(x) = 0 = O(x). If x ∈ dom γ,
then we conclude that

((β • γ) • ((α • β) • (α • γ)))(x) = (β • γ)(x) · ((α • β) • (α • γ))(x)

= (β • γ)(x) · ((α • β)(x) · (α • γ)(x))

= (β(x) · γ(x)) · ((α(x) · β(x)) · (α(x) · γ(x)))

= 0

= O(x).

Hence, (β • γ) • ((α • β) • (α • γ)) = O, so (UP-1) is holding.

Let α ∈ P (X) and let x ∈ X.
Case 1: x /∈ domα. Then x ∈ domO − domα. Thus α(x) and (O • α)(x) are not

defined.
Case 2: x ∈ domα. Then x ∈ domO ∩ domα. Thus (O • α)(x) = O(x) · α(x) =

0 · α(x) = α(x).
Hence, O • α = α, so (UP-2) is holding.

Let α ∈ P (X) and let x ∈ X.
Case 1: x /∈ domα. Then (α •O)(x) = 0 = O(x).
Case 2: x ∈ domα. Then x ∈ domα ∩ domO. Thus (α • O)(x) = α(x) · O(x) =

α(x) · 0 = 0 = O(x).
Hence, α •O = O, so (UP-3) is holding.

Let α, β ∈ P (X) be such that α • β = O and β • α = O. Let x ∈ X. Then
(α • β)(x) = O(x) = 0 and (β • α)(x) = O(x) = 0. If x ∈ domα− domβ, then (α • β)(x)
is not defined which is a contradiction. If x ∈ domβ − domα, then (β • α)(x) is not
defined which is a contradiction. If x ∈ domα ∩ domβ, then 0 = (α • β)(x) = α(x) · β(x)
and 0 = (β • α)(x) = β(x) · α(x). By (UP-4), we have α(x) = β(x). If x /∈ domα and
x /∈ domβ, then α(x) and β(x) are not defined. Hence, α = β, so (UP-4) is holding.

Therefore, (P (X), •, O) is a UP-algebra.

Theorem 3. T (X) is a UP-ideal of P (X) and we shall call it the full transformation
UP-algebra induced by a UP-algebra X.



REFERENCES 881

Proof. Clearly, O ∈ T (X). Let α, β, γ ∈ P (X) be such that α • (β • γ) ∈ T (X)
and β ∈ T (X). Then dom (α • (β • γ)) = X and domβ = X and so by (2.1), X =
dom (α • (β • γ)) = (domα − dom (β • γ))

′
. Thus domα − dom (β • γ) = ∅ and so by

(2.1), ∅ = domα − dom (β • γ) = domα − (domβ − dom γ)
′

= domα − (X − dom γ)
′

=
domα−((dom γ)

′
)
′

= domα−dom γ. By (2.1), dom (α•γ) = (domα−dom γ)
′

= ∅′ = X.
That is, α • γ ∈ T (X). Hence, T (X) is a UP-ideal of P (X) and also a UP-filter and a
UP-subalgebra.

Acknowledgements

The authors wish to express their sincere thanks to the referees for the valuable sug-
gestions which lead to an improvement of this paper.

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