Introducing fully UP-semigroups

In this paper, we introduce some new classes of algebras related to UP-algebras and semigroups, called a left UP-semigroup, a right UP-semigroup, a fully UP-semigroup, a left-left UP-semigroup, a right-left UP-semigroup, a left-right UP-semigroup, a right-right UP-semigroup, a fully-left UP-semigroup, a fully-right UP-semigroup, a left-fully UP-semigroup, a right-fully UP-semigroup, a fully-fully UP-semigroup, and find their examples.


Introduction and Preliminaries
In the literature, several researches introduced a new class of algebras related to logical algebras and semigroups such as: In 1993, Jun, Hong and Roh [5] introduced the notion of BCI-semigroups. In 1998, Jun, Xin and Roh [6] renamed the BCI-semigroup as the IS-algebra. In 2006, Kim [7] introduced the notion of KS-semigroups. In 2011, Ahn and Kim [1] introduced the notion of BE-semigroups. In 2015, Endam and Vilela [2] introduced the notion of JB-semigroups. In 2016, Sultana and Chaudhary [8] introduced the notion of BCH-semigroups. In this paper, we introduce some new classes of algebras related to UP-algebras and semigroups, called a left UP-semigroup, a right UP-semigroup, a fully UPsemigroup, a left-left UP-semigroup, a right-left UP-semigroup, a left-right UP-semigroup, a right-right UP-semigroup, a fully-left UP-semigroup, a fully-right UP-semigroup, a leftfully UP-semigroup, a right-fully UP-semigroup, a fully-fully UP-semigroup, and find their examples.
Before we begin our study, we will introduce the definition of a UP-algebra.
Definition 1.1. [3] An algebra A = (A, ·, 0) of type (2, 0) is called a UP-algebra, where A is a nonempty set, · is a binary operation on A, and 0 is a fixed element of A (i.e., a nullary operation) if it satisfies the following axioms: for any x, y, z ∈ A, In a UP-algebra A = (A, ·, 0), the following assertions are valid (see [3,4]).
Definition 1.2. Let A be a nonempty set, · and * are binary operations on A, and 0 is a fixed element of A (i.e., a nullary operation). An algebra A = (A, ·, * , 0) of type (2, 2, 0) in which (A, ·, 0) is a UP-algebra and (A, * ) is a semigroup is called (1) a left UP-semigroup (in short, an l-UP-semigroup) if the operation " * " is left distributive over the operation "·", (2) a right UP-semigroup (in short, an r-UP-semigroup) if the operation " * " is right distributive over the operation "·", (4) a left-left UP-semigroup (in short, an (l, l)-UP-semigroup) if the operation "·" is left distributive over the operation " * " and the operation " * " is left distributive over the operation "·", (5) a right-left UP-semigroup (in short, an (r, l)-UP-semigroup) if the operation "·" is right distributive over the operation " * " and the operation " * " is left distributive over the operation "·", (6) a left-right UP-semigroup (in short, an (l, r)-UP-semigroup) if the operation "·" is left distributive over the operation " * " and the operation " * " is right distributive over the operation "·", (7) a right-right UP-semigroup (in short, an (r, r)-UP-semigroup) if the operation "·" is right distributive over the operation " * " and the operation " * " is right distributive over the operation "·", (8) a fully-left UP-semigroup (in short, an (f, l)-UP-semigroup) if the operation "·" is distributive (on both sides) over the operation " * " and the operation " * " is left distributive over the operation "·", (9) a fully-right UP-semigroup (in short, an (f, r)-UP-semigroup) if the operation "·" is distributive (on both sides) over the operation " * " and the operation " * " is right distributive over the operation "·", (10) a left-fully UP-semigroup (in short, an (l, f )-UP-semigroup) if the operation "·" is left distributive over the operation " * " and the operation " * " is distributive (on both sides) over the operation "·", (11) a right-fully UP-semigroup (in short, an (r, f )-UP-semigroup) if the operation "·" is right distributive over the operation " * " and the operation " * " is distributive (on both sides) over the operation "·", and (12) a fully-fully UP-semigroup (in short, an (f, f )-UP-semigroup) if the operation "·" is distributive (on both sides) over the operation " * " and the operation " * " is distributive (on both sides) over the operation "·".
In what follows, let A and B denote UP-algebras unless otherwise specified. The following proposition is very important for the study of UP-algebras.

Proposition 1.3. (The operations of a UP-algebra P(X) is left distributive over the operations of a semigroup P(X)
) Let X be a universal set. Then the following properties hold: for any A, B, C ∈ P(X),

Proposition 1.5. (The operations of a semigroup P(X) is left distributive over the operations of a UP-algebra P(X)
) Let X be a universal set. Then the following properties hold: for any A, B, C ∈ P(X), Proposition 1.6. (The operations of a semigroup P(X) is right distributive over the operations of a UP-algebra P(X)) Let X be a universal set. Then the following properties hold: for any A, B, C ∈ P(X), Proposition 1.7. Let X be a universal set. Then the following properties hold: for any A, B, C ∈ P(X),  Proof.

Conclusion
We have introduced the notions of left UP-semigroups, right UP-semigroups, fully UPsemigroups, left-left UP-semigroups, right-left UP-semigroups, left-right UP-semigroups, right-right UP-semigroups, fully-left UP-semigroups, fully-right UP-semigroups, left-fully UP-semigroups, right-fully UP-semigroups and fully-fully UP-semigroups, and have found examples. We have that right-fully UP-semigroups and fully-fully UP-semigroups coincide, and it is only {0}. In further study, we will apply the notion of fuzzy sets and fuzzy soft sets to the theory of all above notions.