mathematics Article Prime i-Ideals in Ordered n-ary Semigroups Patchara Pornsurat , Pakorn Palakawong na Ayutthaya and Bundit Pibaljommee * Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand; [email protected] (P.P.); [email protected] (P.P.n.A.) * Correspondence: [email protected] Abstract: We study the concept of i-ideal of an ordered n-ary semigroup and give a construction of the i-ideal of an ordered n-ary semigroup generated by its nonempty subset. Moreover, we study the notions of prime, weakly prime, semiprime and weakly semiprime ideals of an ordered n-ary semigroup. Keywords: n-ary semigroup; ordered n-ary semigroup; prime ideal; weakly prime ideal; semiprime ideal 1. Introduction A semigroup, which is a well-known generalization of groups, is an algebraic structure appearing in a natural manner in some applications concerning the theory of automata formal languages and other branches of applied mathematics (for example, see [1–5]). A notable particular kind of semigroup is a semigroup together with a partially ordered relation, namely an ordered semigroup. Obviously, we are always able to see a semigroup f( ) 2 × g Citation: Pornsurat, P.; S as an ordered semigroup together with the equality relation x, x S S . Some Palakawong na Ayutthaya, P.; classical and notable researches on ordered semigroups are [6–9]. Additionally, Kehayopulu Pibaljommee, B. Prime i-Ideals in and Tsingelis defined several types of regularities on ordered semigroups in [10,11]. Ordered n-ary Semigroups. A ternary algebraic system was first introduced as a certain ternary algebraic system Mathematics 2021, 9, 491. called a triplex, which turns out to be a ternary group by Lehmer [12] in 1932. As a https://doi.org/10.3390/ generalization of a ternary group, the notion of a ternary semigroup was known to S. math9050491 Banach. He showed that it is always possible to construct a ternary semigroup from a (binary) semigroup and gave an example in which a ternary semigroup does not necessarily Academic Editor: Irina Cristea reduce to a semigroup. Later, Santiago [13] mainly investigated the notions of an ideal and a bi-ideal of a ternary semigroup and used them to characterize a regular ternary Received: 1 February 2021 semigroup. In 2012, Daddi and Pawar [14] defined the notions of ordered quasi-ideals Accepted: 25 February 2021 and ordered bi-ideals of ordered ternary semigroups, which are special kinds of ternary Published: 27 February 2021 semigroups and also presented regular ordered ternary semigroups in terms of several ideal- theoretical characterizations. Another type of regularity on ordered ternary semigroups, Publisher’s Note: MDPI stays neutral namely an intra-regular ordered ternary semigroup, was introduced and characterized by with regard to jurisdictional claims in S. Lekkoksung and N. Lekkoksung in [15]. Several kinds of regularities of ordered ternary published maps and institutional affil- semigroups were also investigated and characterized in terms of many kinds of ordered iations. ideals by Daddi in [16] and Pornsurat and Pibaljommee in [17]. A generalization of classical algebraic structures to n-ary structures was first intro- duced by Kasner [18] in 1904. In 1928, Dörnte [19] introduced and studied the notion of n-ary groups, which is a generalization of that one of groups. Later, Sioson [20] introduced Copyright: © 2021 by the authors. a regular n-ary semigroup, which is an important generalization of a (binary) semigroup Licensee MDPI, Basel, Switzerland. and verified its properties. In [21], Dudek and Gro´zdzi´nskainvestigated the nature of regu- This article is an open access article lar n-ary semigroups. Moreover, Dudek proved several results and gave many examples of distributed under the terms and n-ary groups in [22,23]. Furthermore, in [24], Dudek also studied potent elements of n-ary conditions of the Creative Commons semigroups (n ≥ 3) and investigated properties of ideals in which all elements are potent. Attribution (CC BY) license (https:// Prime, semiprime, weakly prime and weakly semiprime properties are interesting creativecommons.org/licenses/by/ properties of ideals of semigroups and ordered semigroups. In 1970, Szász [25] showed 4.0/). Mathematics 2021, 9, 491. https://doi.org/10.3390/math9050491 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 491 2 of 11 that a semigroup is intra-regular and the set of all ideals forms a chain if and only if every ideal is prime. Later, Kehayopulu [26] gave the same result in ordered semigroups. Furthermore, Kehayopulu also generalized her work on ordered semigroups [26] to ordered hypersemigroups in [27]. The concepts of prime and semiprime ideals were also studied on ternary semigroups by Shabir and Bashir [28] in 2009. In cases of quasi-ideals and bi-ideals of ternary semigroups, Shabir and Bano [29] introduced the notions of prime, semiprime and strongly prime properties of quasi-ideals and bi-ideals and also characterized ternary semigroups for which each bi-ideal is strongly prime. In this work, we investigate some algebraic properties of an ordered n-ary semigroup, which is an n-ary semigroup together with a partially ordered relation and also a general- ization of ordered semigroups. We also study the concept of an i-ideal of an ordered n-ary semigroup and give the construction of an i-ideal of an ordered n-ary semigroup generated by its nonempty subset. Then we generalize the concept of an i-ideal of an ordered n-ary semigroup to the concept of a L-ideal in order that an i-ideal is a special case of a L-ideal where L = fig. Finally, we study the notions of prime, weakly prime, semiprime and weakly semiprime properties of a L-ideal where L = f1, ng and use them to generalize some results in [26] to an ordered n-ary semigroup. 2. Preliminaries Let N be the set of all natural numbers and i, j, n 2 N. A nonempty set S together with an n-ary operation given by f : Sn ! S, where n ≥ 2, is called an n-ary groupoid [24]. For j 1 ≤ i < j ≤ n, the sequence xi, xi+1, xi+2, ... , xj of elements of S is denoted by xi and if j−i+1 j j xi = xi+1 = ... = xj = x, we write x instead of xi . For j < i, we denote xi as empty 0 symbol and so is x. So, the term f (x1,..., xi, x, x,..., x, xi+j+1,..., xn) | {z } j terms is able to be simply represented by j i n f (x1, x, xi+j+1). The associative law [23] for the n-ary operation f on S is defined by i−1 n+i−1 2n−1 j−1 n+j−1 2n−1 f (x1 , f (xi ), xn+i ) = f (x1 , f (xj ), xn+j ) for all 1 ≤ i ≤ j ≤ n and x1, ... , x2n−1 2 S. An n-ary groupoid (S, f ) is called n-ary semigroup if the n-ary operation f satisfies the associative law. By the associativity of an n-ary semigroup (S, f ), we define the map fk where k ≥ 2 by ( k(n−1)+1) = ( ( ( ( n) 2n−1) ) k(n−1)+1 ) fk x1 f f ..., f f x1 , xn+1 ,... , x(k−1)(n−1)+2 | {z } k terms 2n−1 n 2n−1 for any x1, ... , xk(n−1)+1 2 S. Hence, we note that f2(x1 ) = f ( f (x1 ), xn+1 ) for any x1,..., x2n−1 2 S. Many notations above can also be found in [21–24]. For 1 ≤ i < j ≤ n, the sequence Ai, Ai+1, ... , Aj of nonempty sets of S is denoted by j n the symbol Ai. For nonempty subsets A1 of S, we denote n n f (A1 ) = f f (a1 ) j ai 2 Ai where 1 ≤ i ≤ ng. Mathematics 2021, 9, 491 3 of 11 n n If A1 = A2 = ... = An = A , we write f (A) instead of f (A1 ). If A1 = fa1g, then we n n write f (a1, A2 ) instead of f (fa1g, A2 ), and similarly in other cases. For j < i , we set the 0 i i 0 notations Aj and A to be the empty symbols as a similar way of the notations xj and x. An ordered semigroup is an algebraic structure (S, ·, ≤) such that (S, ·) is a semigroup and (S, ≤) is a poset satisfying the compatibility property, i.e., if a ≤ b, then ac ≤ bc and ca ≤ cb for all a, b, c 2 S. To generalize the notion of n-ary semigroups and ordered (binary) semigroups, the notion of an ordered n-ary semigroup is defined as follows. Definition 1. An ordered n-ary semigroup is a system (S, f , ≤) such that (S, f ) is an n-ary semi- group and (S, ≤) is a partially ordered set satisfying the following property. For any a, b, x1,..., xn 2 S, if a ≤ b, then i−1 n i−1 n f (x1 , a, xi+1) ≤ f (x1 , b, xi+1) for all 1 ≤ i ≤ n. Throughout this paper, we write S for an ordered n-ary semigroup, unless specified otherwise. For any Æ 6= A ⊆ S, we denote (A] = fx 2 S j x ≤ a for some a 2 Ag. Now, we give some basic properties of the operator (] on an ordered n-ary semigroup as follows. Lemma 1. Let A, B, A1 ..., An be nonempty subsets of S. Then the following statements hold: (i) A ⊆ (A]; (ii) (A] = ((A]]; (iii) A ⊆ B implies (A] ⊆ (B]; n (iv) f ((A1], (A2],..., (An]) ⊆ f ((A1 ]); (v) (A [ B] = (A] [ (B]; (vi) (A \ B] ⊆ (A] \ (B]. The following remark is directly obtained by the notion of fk where k = m and k = m + 1.