Residuated operators and Dedekind-MacNeille completion (Q2658302)

Given a bounded partially ordered set (\textit{poset} for short) \(\mathbf{P}=(P, \leqslant, {}^{\prime}, 0, 1)\) equipped with a unary operation \({}^{\prime}\), for every subset \(M\subseteq P\), one can define the following two sets: \(U(M):=\{x\in P\mid y\leq x\) for every \(y\in M\}\) (\textit{upper cone of \(M\)}) and \(L(M):=\{x\in P\mid x\leq y\) for every \(y\in M\}\) (\textit{lower cone of \(M\)}). If \(M=\{x, y\}\) or \(M=\{x\}\), then one writes \(U(x, y)\), \(L(x, y)\) or \(U(x)\), \(L(x)\), respectively. An \textit{operator left residuated poset} is a septuple \(\mathbf{P}=(P, \leqslant, {}^{\prime}, M, R, 0, 1)\), where \((P, \leq, {}^{\prime}, 0, 1)\) is a bounded poset with a unary operation, and \(M,R:P\times P\rightarrow\mathcal{P}(P)\) are maps (\(\mathcal{P}(P)\) stands for the powerset of \(P\)) such that the following three conditions are fulfilled for every \(x,y,z\in P\): \begin{itemize} \item[(1)] \(M(x, 1)=M(1,x)=L(x)\); \item[(2)] \(M(x,y)\subseteq L(z)\) if and only if \(L(x)\subseteq R(y,z)\); and \item[(3)] \(R(x,0)=L(x^{\prime})\). \end{itemize} In the present paper, the authors work with posets \(\mathbf{P}=(P, \leq, {}^{\prime}, 0, 1)\), where \({}^{\prime}\) is an antitone involution or a complementation (defined appropriately). A \textit{left residuated lattice} is an algebra \(\mathbf{L} = (L,\vee,\wedge,\odot, \rightarrow,1)\) having type \((2,2,2,2,0)\) such that \((L,\vee,\wedge, 1)\) is a lattice with the greatest element \(1\), and, additionally, the following two conditions hold for every \(x,y,z\in L\): \begin{itemize} \item[(4)] \(x\odot 1=x=1\odot x\); and \item[(5)] \(x\odot y\leq z\) if and only if \(x\leq y\rightarrow z\). \end{itemize} Every poset \(\mathbf{P}=(P,\leq)\) has its Dedekind-MacNeille completion \(\mathbf{DM}(\mathbf{P})\), which is a complete lattice. The present authors say that an expression in the above operators \(U\) and \(L\) is \(\mathbf{DM}\)-\textit{transformed} provided that every expression \(U(x,y)\) or \(LU(x,y)\) is substituted by \(x\vee y\), and every expression \(L(x,y)\) is substituted by \(x\wedge y\). For example, the expression \(L(U(x,y),U(x,z))\) is written as \((x\vee y)\wedge(x\vee z)\), and the expression \(LU(x,L(y,z))\) is written as \(x\vee(y\wedge z)\). The aim of the present paper is as follows. Having an operator left residuated poset \(\mathbf{P}=(P, \leq, {}^{\prime}, M, R, 0\), \(1)\), the authors ask, whether the operators \(M\) and \(R\) expressed in \(U\) and \(L\) can be \(\mathbf{DM}\)-transformed in such a way that the resulting expressions will be binary operations \(\odot\) and \(\rightarrow\) on \(\mathbf{DM}(\mathbf{P})\) satisfying the above properties \((4)\) and \((5)\) in the Dedekind-MacNeille completion \(\mathbf{DM}(\mathbf{P})\) of \textbf{P}. The paper is well written, gives most of its required preliminaries (the omitted ones can be found in the carefully chosen references at the end of the paper), and will be of interest to researchers studying algebraic aspects of posets. For the entire collection see [Zbl 1459.06001].