23,915 research outputs found
The lattice of embedded subsets
In cooperative game theory, games in partition function form are real-valued function on the set of so-called embedded coalitions, that is, pairs where is a subset (coalition) of the set of players, and is a partition of containing . Despite the fact that many studies have been devoted to such games, surprisingly nobody clearly defined a structure (i.e., an order) on embedded coalitions, resulting in scattered and divergent works, lacking unification and proper analysis. The aim of the paper is to fill this gap, thus to study the structure of embedded coalitions (called here embedded subsets), and the properties of games in partition function form.Partition; Embedded subset; Game; Valuation; k-monotonicity
Sublattices of complete lattices with continuity conditions
Various embedding problems of lattices into complete lattices are solved. We
prove that for any join-semilattice S with the minimal join-cover refinement
property, the ideal lattice IdS of S is both algebraic and dually algebraic.
Furthermore, if there are no infinite D-sequences in J(S), then IdS can be
embedded into a direct product of finite lower bounded lattices. We also find a
system of infinitary identities that characterize sublattices of complete,
lower continuous, and join-semidistributive lattices. These conditions are
satisfied by any (not necessarily finitely generated) lower bounded lattice and
by any locally finite, join-semidistributive lattice. Furthermore, they imply
M. Ern\'e's dual staircase distributivity. On the other hand, we prove that the
subspace lattice of any infinite-dimensional vector space cannot be embedded
into any countably complete, countably upper continuous, and countably lower
continuous lattice. A similar result holds for the lattice of all order-convex
subsets of any infinite chain.Comment: To appear in Algebra Universali
Sublattices of lattices of convex subsets of vector spaces
For a left vector space V over a totally ordered division ring F, let Co(V)
denote the lattice of convex subsets of V. We prove that every lattice L can be
embedded into Co(V) for some left F-vector space V. Furthermore, if L is finite
lower bounded, then V can be taken finite-dimensional, and L embeds into a
finite lower bounded lattice of the form ,
for some finite subset of . In particular, we obtain a new universal
class for finite lower bounded lattices
Partial orderings for hesitant fuzzy sets
New partial orderings =o=o, =p=p and =H=H are defined, studied and compared on the set HH of finite subsets of the unit interval with special emphasis on the last one. Since comparing two sets of the same cardinality is a simple issue, the idea for comparing two sets A and B of different cardinalities n and m respectively using =H=H is repeating their elements in order to obtain two series with the same length. If lcm(n,m)lcm(n,m) is the least common multiple of n and m we can repeat every element of A lcm(n,m)/mlcm(n,m)/m times and every element of B lcm(n,m)/nlcm(n,m)/n times to obtain such series and compare them (Definition 2.2).
(H,=H)(H,=H) is a bounded partially ordered set but not a lattice. Nevertheless, it will be shown that some interesting subsets of (H,=H)(H,=H) have a lattice structure. Moreover in the set BB of finite bags or multisets (i.e. allowing repetition of objects) of the unit interval a preorder =B=B can be defined in a similar way as =H=H in HH and considering the quotient set View the MathML sourceB¿=B/~ of BB by the equivalence relation ~ defined by A~BA~B when A=BBA=BB and B=BAB=BA, View the MathML source(B¿,=B) is a lattice and (H,=H)(H,=H) can be naturally embedded into it.Peer ReviewedPostprint (author's final draft
Sublattices of lattices of order-convex sets, I. The main representation theorem
For a partially ordered set P, we denote by Co(P) the lattice of order-convex
subsets of P. We find three new lattice identities, (S), (U), and (B), such
that the following result holds. Theorem. Let L be a lattice. Then L embeds
into some lattice of the form Co(P) iff L satisfies (S), (U), and (B).
Furthermore, if L has an embedding into some Co(P), then it has such an
embedding that preserves the existing bounds. If L is finite, then one can take
P finite, of cardinality at most , where n is the number of
join-irreducible elements of L. On the other hand, the partially ordered set P
can be chosen in such a way that there are no infinite bounded chains in P and
the undirected graph of the predecessor relation of P is a tree
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