Structures of compactly generated lattices described by cut sets of
· Xue-ping Wang
· Xiaohong Zhang
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
This paper describes the distributivity, the modularity, the semimodularity and the lower semimodularity of compactly gener-
ated lattices from a view of cut sets of L-valued sets, respectively. Applying terms of cut sets of L-valued sets, it gives some
sufﬁcient and necessary conditions which can be used to determine whether a compactly generated lattice is distributive,
modular, semimodular and lower semimodular, respectively.
Keywords L-valued set · Cut set · Compactly generated lattice · Distributive lattice · Modular lattice · Semimodular lattice ·
Lower semimodular lattice
Lattice-valued mathematics had undergone a signiﬁcant
development from Goguen’s (1967) ﬁrst paper. In this way,
bridges had been created between fuzzy mathematics and
other branches like automata and tree series (Borchardt
et al. 2006), theoretical computer science (Chechik et al.
2001), fuzzy control systems (Bˇelohlávek 2002;Bˇelohlávek
and Vychodil 2005) and fuzzy congruences (Šešelja and
Tepavˇcevi´c 1994). It is known that the collection of cut sets
of a lattice-valued set (an L-valued set for short) can be
considered as an ordered structure. More importantly, it is
Communicated by A. Di Nola.
Department of Mathematics, Chengdu Normal University,
Chengdu 611130, People’s Republic of China
College of Mathematics and Software Science, Sichuan
Normal University, Chengdu 610066, People’s Republic of
College of Arts and Sciences, Shanghai Maritime University,
Shanghai 201306, People’s Republic of China
possible to present an ordered structure by cut sets of a suit-
able fuzzy set μ and consequently by the mapping μ itself
as pointed out by Šešelja and Tepavˇcevi´c(2003a,b). This
topic has been discussed by many scholars (see, e.g., Jiménez
et al. 2010, 2011a,b; Gorjanac-Ranitovi´c and Petojevi´c
2014; Gorjanac-Ranitovi´c and Tepavˇcevi´c 2018; Šešelja et al.
2008). On the other hand, the distributivity, the modularity,
the semimodularity and the lower semimodularity of com-
pactly generated lattices are important topics in the theory
of lattices (see, e.g., Crawley and Dilworth 1973; Czédli and
Schmidt 2010; Grätzer and Wehrung 2014; Skublics 2013).
Therefore, a natural problem is whether we can describe
the distributivity, the modularity, the semimodularity and
the lower semimodularity of compactly generated lattices
from a view of cut sets of L-valued sets, respectively. The
aim of this paper is to reveal a role of cut sets of L-
valued sets in describing the distributivity, the modularity and
the semimodularity of compactly generated lattices, respec-
The paper is organized as follows. First, for the sake of
convenience, some notions and previous results are given in
Sect. 2 which are the basis for our investigation. Next, more
connections between compactly generated lattices and the
cut sets of L-valued sets are investigated in Sect. 3. Then
the distributivity, the modularity, the semimodularity and the
lower semimodularity of a compactly generated lattice are
described from a view of cut sets of L-valued sets in Sect. 4.
Finally, a conclusion is drawn in Sect. 5.