Semiprime ideals and separation theorems for posets
Let \(P\) be a poset and let \(A\) be a subset of \(P\). Define \(A^{u}:=\{x\in P : x\geq a \text{ for every } a\in A\}\). Dually define \(A^{l}:=\{x\in P : x\leq a \text{ for every } a\in A\}\). Then \(A^{ul}\) means \(\{A^{u}\}^l\) and \(A^{lu}\) means \(\{A^{l}\}^u\). A subset \(I\) of \(P\) is called an ideal if \(a,b\in I\) implies that \(\{a,b\}^{ul}\subseteq I\). A proper ideal \(I\) is called prime if \(\{a,b\}^{l}\subseteq I\) implies that either \(a\in I\) or \(b\in I\). Dually define filter and prime filter. These notions are due, respectively, to \textit{R. Halaš} [``Characterization of distributive sets by generalized annihilators'', Arch. Math., Brno 30, No.~1, 25--27 (1994; Zbl 0805.06001)] and \textit{R. Halaš} and \textit{J. Rachůnek} [``Polars and prime ideals in ordered sets'', Discuss. Math., Algebra Stoch. Methods 15, No. 1, 43--59 (1995; Zbl 0840.06003)]. In [``The theory of representations for Boolean algebras'', Trans. Am. Math. Soc. 40, 37--111 (1936; Zbl 0014.34002 and JFM 62.0033.04)], \textit{M. H. Stone} proved a separation theorem in the case of distributive lattices: Let \(L\) be a distributive lattice, let \(I\) be an ideal, let \(D\) be a filter of \(L\) such that \(I\cap D=\emptyset\). Then there exists a prime ideal \(P\) of \(L\) such that \(I\subseteq P\) and \(P\cap D=\emptyset\). In [``Semiprime ideals in general lattices'', J. Pure Appl. Algebra 56, No. 2, 105--118 (1989; Zbl 0665.06006)], \textit{Y. Rav} proved an analogue of Stone's Theorem for semiprime ideals in lattices. An ideal \(I\) of a lattice \(L\) is semiprime if \(x\wedge y\in I\) and \(x\wedge z\in I\) together imply \(x\wedge(y\vee z)\). In the paper under review the authors generalize the notion of semiprime ideal in lattices to posets as follows. An ideal \(I\) of a poset \(P\) is semiprime if \(\{a,b\}^{l}\subseteq I\) and \(\{a,c\}^{l}\subseteq I\) together imply \(\{a,\{b,c\}^{u}\}^{l}\subseteq I\). The authors then prove theorems analoguous to Stone's and Rav's in finite posets. The authors observe by means of counterexamples that their results do not extend to infinite posets.
scientific article; zbMATH DE number 5366784