A note on \((m,n)\)-ideals in regular duo ordered semigroups. (Q2798962)

The purpose of this paper is to prove that, for a regular duo ordered semigroup, every \((m,n)\)-ideal is a two-sided ideal (see also the abstract and the introduction). This is given in Theorem 2.5 of the paper.NEWLINENEWLINE This theorem is based on the following Theorem 1.4: Let \((S,.,\leq)\) be an ordered semigroup. Then a subset \(A\) of \(S\) is a \(\pi\)-ideal of \(S\) if and only if it is an \((m,n)\)-ideal of \(S\). The concept of the \(\pi\)-ideal in it is not clear. The \(\pi\)-ideal in the theorem should be defined in terms of \(m\) and \(n\), and it is strange that the authors included the Corollary 1.3 in the paper. In Theorem 1.4, they should have said directly, that a subset \(A\) of \(S\) is an \(r^ml^n\)-ideal of \(S\) if and only if \(A\) is an \((m,n)\)-ideal of \(S\) (and delete the Corollary 1.3). But the concept of \(r^ml^n\)-ideal is also not clear. In spite of that, in the proof of the main theorem (Theorem 2.5), several times, they write: \(L_i\) (\(i=1,2,\ldots,k\)), for example, and then they write \(L_0=S\). They write \(R_j\) (\(j=1,2,\dots,k+1\)), and then \(R_0=S\). Anyway, the main theorem is based on unclear results, in addition there are mistakes in its proof.NEWLINENEWLINE Some further remarks: According to the authors, Corollary 1.3 is a consequence of Theorem 1.2. If it is so, then a proof is necessary for Corollary 1.3, it is not obvious at all. The main theorem is also based on Theorems 1.1 and 2.4. First of all we have to mention that, in a regular ordered semigroup, a nonempty subset \(B\) has the property \(BSB\subseteq B\) if and only if it has the properties \(BSB\subseteq B\) and \(B^2\subseteq B\). This is Theorem 1.1 of the paper for which they refer to the paper by \textit{T. Changphas} [Far East J. Math. Sci. (FJMS) 79, No. 2, 177-183 (2013; Zbl 1305.06009)] = [3] of the References published in 2013: A nonempty subset \(A\) of a regular ordered semigroup \(S\) is a bi-ideal of \(S\) if and only if there exist a left ideal \(L\) and a right ideal \(R\) of \(S\) such that \(A=(RL]\). This theorem is Theorem 1 in the paper [\textit{N. Kehayopulu}, PU.M.A., Pure Math. Appl. 6, No. 4, 333-344 (1995; Zbl 0854.06023)].NEWLINENEWLINE This is Theorem 2.4: If \(S\) is a regular duo ordered semigroup, then every bi-ideal of \(S\) is a two-sided ideal of \(S\). For this theorem they also refer to [3] in the References, while this is the Lemma 3 in [Zbl 0854.06023, loc. cit.]. For the definition of the bi-ideal of ordered semigroups, the authors refer to [\textit{N. Kehayopulu}, Math. Jap. 37, No. 1, 123-130 (1992; Zbl 0745.06007)] ([7] in the References), while in [7] a bi-ideal is defined as a nonempty subset \(B\) of \(S\) satisfying the properties \(BSB\subseteq B\) and \((B]=B\) and not as a subsemigroup of \(S\) satisfying the same properties (see also the paper by Kehayopulu in 1995 mentioned above, edited by S. Lajos in Pure Math. Appl.). For the definition of \((m,n)\)-ideal the authors refer to the paper by \textit{J. Sanborisoot} and \textit{T. Changphas} [Far East J. Math. Sci. (FJMS) 65, No. 1, 75-86 (2012; Zbl 1284.06039)] published in 2012 ([11] in the References), while the concept of \((m,n)\)-ideal element of a poe-semigroup has been first defined in [\textit{N. Kehayopulu}, Semigroup Forum 25, 213-222 (1982; Zbl 0499.06012)] and this definition has been naturally transferred to a po-semigroup \(S\) as a nonempty subset \(B\) of \(S\) satisfying the properties \(B^mSB^n\subseteq B\) and \((B]=B\) [\textit{N. Kehayopulu, M. Tsingelis}, PU.M.A., Pure Math. Appl. 10, No. 1, 59-67 (1999; Zbl 0933.06007)]. For the concept of regular duo ordered semigroups the authors refer to the paper by Changphas [loc. cit.] [3] in the References, while this definition certainly does not belong to Changphas (see, for example, [\textit{N. Kehayopulu}, Math. Jap. 37, No. 3, 535-540 (1992; Zbl 0760.06006)]).NEWLINENEWLINE For the examples of the paper one needs computer programs, it is impossible to construct them by hand, and there is no reference to such programs in the paper. Of course one can get them from published papers, but in that case these papers should be cited in the References. In addition, the right and the left ideals should be given in the examples to show that the regular ordered semigroups in the examples are duo. Finally, the authors give three examples of regular duo ordered semigroups. As all of them are given by a table of multiplication and an order, one example is enough. Besides, the examples are not connected with the aim of the paper. In short, this paper should be rewritten.