Abstract: In this paper, we explore the behavior with respect to direct and inverse images of the relations between the different generalizations of filter notion studied in [Gutiérrez García J, de Prada Vicente MA. Characteristic values of inverted perpendicular-filters. Fuzzy Sets Syst 2005;56:55–67]. Also, we give sufficient conditions for the existence of an upper bound for those generalizations of filters.

Abstract: Given a complete lattice $(L,le)$ equipped with an order-re-ver-sing involution $'$ (also called a complete De Morgan algebra), we find conditions for the existence of a residuated binary operation $ast$ on $L$ such that the given order-reversing involution is determined by the residuation associated to $ast$. As a consequence, in the case of completely distributive lattices with an order-reversing involution, we find a necessary and sufficient condition for the desired residuated binary operation $ast$ to exist.

Abstract: The concepts of upper and lower semicontinuity in pointfree topology were introduced and first studied by Li and Wang [Y.-M. Li, G.-J. Wang, Localic Katětov–Tong insertion theorem and localic Tietze extension theorem, Comment. Math. Univ. Carolin. 38 (1997) 801–814]. However Li and Wang’s treatment does not faithfully reflect the original classical notion. In this note, we present algebraic descriptions of upper and lower semicontinuous real functions, in terms of frame homomorphisms, that suggest the right alternative to the definitions of Li and Wang. This fixes the discrepancy between the classical and the pointfree notions and turns out to be the appropriate notion that makes the Katětov–Tong theorem provable in the pointfree context without any restrictions.

Abstract: This paper deals with completeness of Hutton <a name="mml2"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4NCSGHW-2&_mathId=mml2&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=60059fada17e3e0ef12c1adf2cbd3551" title="Click to view the MathML source">[0,1]</a>-quasi-uniform spaces. Recently, the first two authors, [J. Guti&#xe9;rrez Garc&#xed;a, M.A. de Prada Vicente, Hutton <a name="mml3"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4NCSGHW-2&_mathId=mml3&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=a45d4ea47f40bf8971abf71815b2a8c1" title="Click to view the MathML source">[0,1]</a>-quasi-uniformities induced by fuzzy (quasi-)metric spaces, Fuzzy Sets and Systems 157 (2006), 755&#x2013;766], have constructed a Hutton <a name="mml4"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4NCSGHW-2&_mathId=mml4&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=ac139be62de11ef48d71b5c21c9809fd" title="Click to view the MathML source">[0,1]</a>-quasi-uniformity induced by a fuzzy metric space (in the sense of George and Veeramani). In this paper, we define completeness of Hutton <a name="mml5"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4NCSGHW-2&_mathId=mml5&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=9acd095b4999cfd0e18602252ab64125" title="Click to view the MathML source">[0,1]</a>-quasi-uniform spaces as convergence of any stratified tight Cauchy <a name="mml6"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4NCSGHW-2&_mathId=mml6&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=48f64d1d19fb302df86f62c34f5e6487" title="Click to view the MathML source">[0,1]</a>-filter. Our main result states the equivalence between completeness of any fuzzy metric space <a name="mml7"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4NCSGHW-2&_mathId=mml7&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=eb18f7420cdb2f7a69796db5de97b880" title="Click to view the MathML source">(<i>X</i>,<i>M</i>,*)</a> and completeness of the induced Hutton <a name="mml8"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4NCSGHW-2&_mathId=mml8&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=a06a321491e96105af31d1c79f853a04" title="Click to view the MathML source">[0,1]</a>-quasi-uniformity <a name="mml9"></a><a href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4NCSGHW-2&_mathId=mml9&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=ca9c8ccfba67f414efcc312d48524e62"><img src="http://www.sciencedirect.com/cache/MiamiImageURL/B6V05-4NCSGHW-2-J/0?wchp=dGLbVzz-zSkWA" alt="Click to view the MathML source" align="absbottom" border="0" height=15 width=25></a>. Also it is proved that the Hutton <a name="mml10"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4NCSGHW-2&_mathId=mml10&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=8d6601bf30a6e53d8c12890dd0e7ad57" title="Click to view the MathML source">[0,1]</a>-quasi-uniform space <a name="mml11"></a><a href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4NCSGHW-2&_mathId=mml11&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=71814b19259bb1a0e3d9f83cfa3b0a65"><img src="http://www.sciencedirect.com/cache/MiamiImageURL/B6V05-4NCSGHW-2-3/0?wchp=dGLbVzz-zSkWA" alt="Click to view the MathML source" align="absbottom" border="0" height=15 width=59></a> has, in this context, a kind of completion that is unique up to uniform isomorphism. The obtained results come from an appropriate definition of Cauchy <i>L</i>-filter (where <i>L</i> stands for a complete lattice, with additional properties).

Abstract: Problems of inserting lattice-valued functions are investigated. We provide an analogue of the classical insertion theorem of Lane [Proc. Amer. Math. Soc. 49 (1975) 90&#x2013;94] for <i>L</i>-valued functions where <i>L</i> is a <img src="http://www.sciencedirect.com/scidirimg/entities/22b2.gif" alt="left triangle, open, var" border=0>-separable completely distributive lattice (i.e. <i>L</i> admits a countable join-dense subset which is free of completely join-irreducible elements). As a corollary we get an <i>L</i>-version of the Kat&#x11b;tov&#x2013;Tong insertion theorem due to Liu and Luo [Topology Appl. 45 (1992) 173&#x2013;188] (our proof is different and much simpler). We show that <img src="http://www.sciencedirect.com/scidirimg/entities/22b2.gif" alt="left triangle, open, var" border=0>-separable completely distributive lattices are closed under the formation of countable products. In particular, the Hilbert cube is a <img src="http://www.sciencedirect.com/scidirimg/entities/22b2.gif" alt="left triangle, open, var" border=0>-separable completely distributive lattice and some join-dense subset is shown to be both order and topologically isomorphic to the hedgehog <a name="mml1"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V1K-4GNKR4J-1&_mathId=mml1&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=5fe5a10f8a5c72224328f78099826d5a" title="Click to view the MathML source"><i>J</i>(<i>&#x3c9;</i>)</a> with appropriately defined topology. This done, we deduce an insertion theorem for <a name="mml2"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V1K-4GNKR4J-1&_mathId=mml2&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=a206ba55a9494beed189ab686d55d532" title="Click to view the MathML source"><i>J</i>(<i>&#x3c9;</i>)</a>-valued functions which is independent of that of Blair and Swardson [Indian J. Math. 29 (1987) 229&#x2013;250]. Also, we provide an iff criterion for inserting a pair of semicontinuous function which yields, among others, a characterization of hereditarily normal spaces.

Abstract: It is well known that given a probabilistic metric space (Menger space) with continuous t-norm <a name="mml2"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4HR71HX-1&_mathId=mml2&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=a6cbaae68995dd48c9e93d66b18424e5" title="Click to view the MathML source"><i>T</i></a> there is a Hausdorff topology associated. This association factorizes through strong uniformities (or <a name="mml3"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4HR71HX-1&_mathId=mml3&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=54a162951594fbb41ea05a9a76469783" title="Click to view the MathML source">(<i>&#x3b5;</i>,<i>&#x3bb;</i>)</a>-uniformities). Similarly, any fuzzy metric space <a name="mml4"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4HR71HX-1&_mathId=mml4&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=031a8c25e691708b4dcab1112e912dce" title="Click to view the MathML source">(<i>X</i>,<i>M</i>,*)</a> can be endowed with a Hausdorff topology <a name="mml5"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4HR71HX-1&_mathId=mml5&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=9ce79792d2ab6a96352c6ecc2a49e64b" title="Click to view the MathML source"><i>&#x3c4;</i>_<i>M</i></a> (in the case of fuzzy quasi-metric spaces, a <a name="mml6"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4HR71HX-1&_mathId=mml6&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=139ef3535a19fee4e73acba509f3f1b2" title="Click to view the MathML source"><i>T</i>₁</a> topology), and again this association factorizes through (quasi-)uniform spaces. In this paper we associate to each fuzzy (quasi-)metric space a Hutton <a name="mml7"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4HR71HX-1&_mathId=mml7&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=2eba8260a62518f4fd5bf1d3666bed43" title="Click to view the MathML source">[0,1]</a>-quasi-uniformity <a name="mml8"></a><a href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4HR71HX-1&_mathId=mml8&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=70e85a084138d04001d39bc1dae44e76"><img src="http://www.sciencedirect.com/cache/MiamiImageURL/B6V05-4HR71HX-1-H/0?wchp=dGLbVtb-zSkWA" alt="Click to view the MathML source" align="absbottom" border="0" height=14 width=27></a>. This allows us to give a factorization of the previous association via Hutton <a name="mml9"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4HR71HX-1&_mathId=mml9&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=a905002c134fea0671a137c57d2ee950" title="Click to view the MathML source">[0,1]</a>-quasi-uniformities. It is also proved that the topology <a name="mml10"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4HR71HX-1&_mathId=mml10&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=97461d401c1b2c76f57ccbbd7d5ea906" title="Click to view the MathML source"><i>&#x3c4;</i>_<i>M</i></a> is exactly the image under Lowen's functor <a name="mml11"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4HR71HX-1&_mathId=mml11&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=881f0c97fc9b7ec7ba028efff225283d" title="Click to view the MathML source"><i>&#x3b9;</i></a> of the <a name="mml12"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4HR71HX-1&_mathId=mml12&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=d680d0bfe52e00ed36f3625d1d2749fd" title="Click to view the MathML source">[0,1]</a>-topology induced by <a name="mml13"></a><a href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4HR71HX-1&_mathId=mml13&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=c8e9de095a9e4c1df3b7ab1f93a0e855"><img src="http://www.sciencedirect.com/cache/MiamiImageURL/B6V05-4HR71HX-1-5/0?wchp=dGLbVtb-zSkWA" alt="Click to view the MathML source" align="absbottom" border="0" height=14 width=27></a>. As a consequence, we get a class of Hutton <a name="mml14"></a><a style="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V05-4HR71HX-1&_mathId=mml14&_user=984461&_rdoc=1&_acct=C000049823&_version=1&_userid=984461&md5=c2c8639f3d098d630f12a5442e736f10" title="Click to view the MathML source">[0,1]</a>-quasi-uniformities which are probabilistic metrizable.