The Length of Chains in Algebraic Lattices
نویسنده
چکیده
We study how the existence in an algebraic lattice L of a chain of a given type is reflected in the join-semilattice K(L) of its compact elements. We show that for every chain α of size κ, there is a set B of at most 2 join-semilattices, each one having a least element such that an algebraic lattice L contains no chain of order type I(α) if and only if the join-semilattice K(L) of its compact elements contains no join-subsemilattice isomorphic to a member of B. We show that among the join-subsemilattices of [ω] belonging to B, one is embeddable in all the others. We conjecture that if α is countable, there is a finite set B.
منابع مشابه
Algebraic Properties of Intuitionistic Fuzzy Residuated Lattices
In this paper, we investigate more relations between the symmetric residuated lattices $L$ with their corresponding intuitionistic fuzzy residuated lattice $tilde{L}$. It is shown that some algebraic structures of $L$ such as Heyting algebra, Glivenko residuated lattice and strict residuated lattice are preserved for $tilde{L}$. Examples are given for those structures that do not remain the sam...
متن کاملRADICAL OF FILTERS IN RESIDUATED LATTICES
In this paper, the notion of the radical of a filter in residuated lattices is defined and several characterizations of the radical of a filter are given. We show that if F is a positive implicative filter (or obstinate filter), then Rad(F)=F. We proved the extension theorem for radical of filters in residuated lattices. Also, we study the radical of filters in linearly o...
متن کاملState filters in state residuated lattices
In this paper, we introduce the notions of prime state filters, obstinate state filters, and primary state filters in state residuated lattices and study some properties of them. Several characterizations of these state filters are given and the prime state filter theorem is proved. In addition, we investigate the relations between them.
متن کاملFrankl's Conjecture for a subclass of semimodular lattices
In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices ha...
متن کاملAn equivalence functor between local vector lattices and vector lattices
We call a local vector lattice any vector lattice with a distinguished positive strong unit and having exactly one maximal ideal (its radical). We provide a short study of local vector lattices. In this regards, some characterizations of local vector lattices are given. For instance, we prove that a vector lattice with a distinguished strong unit is local if and only if it is clean with non no-...
متن کاملA SHORT NOTE ON ATOMS AND COATOMS IN SUBGROUP LATTICES OF GROUPS
In this paper we give an elementary argument about the atoms and coatoms of the latticeof all subgroups of a group. It is proved that an abelian group of finite exponent is strongly coatomic.
متن کامل