Atomicity of the poset of multifuncoids

OpenYear of origin: 2018

Posted online: 2018-12-01 21:10:30Z by Victor Lvovich Porton22

Cite as: P-181201.13

  • General Topology
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Definition A free star on a join-semilattice $ \mathfrak{A} $ with least element 0 is a set $ S $ such that $ 0 \not\in S $ and $$ \forall A, B \in \mathfrak{A}: \left( A \cup B \in S \Leftrightarrow A \in S \vee B \in S \right) . $$

Definition Let $ \mathfrak{A} $ be a family of posets, $ f \in \mathscr{P} \prod \mathfrak{A} $ ($ \prod \mathfrak{A} $ has the order of function space of posets), $ i \in \operatorname{dom}\mathfrak{A} $, $ L \in \prod \mathfrak{A}|_{\left( \operatorname{dom}\mathfrak{A} \right) \setminus \left\{ i \right\}} $. Then $$ \left( \operatorname{val}f \right)_i L = \left\{ X \in \mathfrak{A}_i \hspace{0.5em} | \hspace{0.5em} L \cup \left\{ (i ; X) \right\} \in f \right\} . $$

Definition Let $ \mathfrak{A} $ is a family of posets. A multidimensional funcoid (or multifuncoid for short) of the form $ \mathfrak{A} $ is an $ f \in \mathscr{P} \prod \mathfrak{A} $ such that we have that:

- $ \left( \operatorname{val} f \right)_i L $ is a free star for every $ i \in \operatorname{dom} \mathfrak{A} $, $ L \in \prod \mathfrak{A}|_{\left( \operatorname{dom} \mathfrak{A} \right) \setminus \left\{ i \right\}} $.

- $ f $ is an upper set.

$ \mathfrak{A}^n $ is a function space over a poset $ \mathfrak{A} $ that is $ a\le b\Leftrightarrow \forall i\in n:a_i\le b_i $ for $ a,b\in\mathfrak{A}^n $.

Definition Let $ \mathfrak{A} $ is a family of join-semilattice. A completary multifuncoid of the form $ \mathfrak{A} $ is an $ f \in \mathscr{P} \prod \mathfrak{A} $ such that we have that:

- $ L_0 \cup L_1 \in f \Leftrightarrow \exists c \in \left\{ 0, 1 \right\}^n : \left( \lambda i \in n : L_{c \left( i_{} \right)} i \right) \in f $ for every $ L_0, L_1 \in \prod \mathfrak{A} $.

- If $ L \in \prod \mathfrak{A} $ and $ L_i = 0^{\mathfrak{A}_i} $ for some $ i $ then $ \neg f L $.

$ \mathfrak{A}^n $ is a function space over a poset $ \mathfrak{A} $ that is $ a\le b\Leftrightarrow \forall i\in n:a_i\le b_i $ for $ a,b\in\mathfrak{A}^n $.

If the below conjectures are solved true, they may probably be generalized for general multifuncoids of more general forms than $(\mathscr{P}\mho)^n$.

Problem's Description

Conjecture The poset of multifuncoids of the form $ (\mathscr{P}\mho)^n $ is for every sets $ \mho $ and $ n $:

- atomic;

- atomistic.

  1. BookIs an originAlgebraic General Topology. Volume 1

    year of publication: 2018fulltext


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  • Created at: 2018-12-01 21:10:30Z