2025-01-12

947: Multilinear Map

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definition of multilinear map

Topics


About: module

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of multilinear map.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description 1


Here is the rules of Structured Description.

Entities:
\( J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\( R\): \(\in \{\text{ the rings }\}\)
\( \{M_j \vert j \in J\}\): \(\subseteq \{\text{ the } R \text{ modules }\}\)
\( \times_{j \in J} M_j\): \(= \text{ the product module }\)
\( M\): \(\in \{\text{ the } R \text{ modules }\}\)
\(*f\): \(: \times_{j \in J} M_j \to M\)
//

Conditions:
\(\forall m \in \times_{j \in J} M_j \text{ such that } m (l) = r' m'_l + r'' m''_l \text{ for any } l \in J, \text{ any } r', r'' \in R, \text{ and any } m'_l, m''_l \in M_l (f (m) = r' f (m') + r'' f(m'') \text{ where } m' \in \times_{j \in J} M_j \text{ is } m' (l) = m'_l \text{ and } m' (j) = m (j) \text{ for each } j \neq l \text{ and } m'' \in \times_{j \in J} M_j \text{ is } m'' (l) = m''_l \text{ and } m'' (j) = m (j) \text{ for each } j \neq l)\)
//


2: Structured Description 2


Here is the rules of Structured Description.

Entities:
\( R\): \(\in \{\text{ the rings }\}\)
\( \{M_1, ..., M_k\}\): \(\subseteq \{\text{ the } R \text{ modules }\}\)
\( M_1 \times ... \times M_k\): \(= \text{ the product module }\)
\( M\): \(\in \{\text{ the } R \text{ modules }\}\)
\(*f\): \(: M_1 \times ... \times M_k \to M\)
//

Conditions:
\(\forall m \in M_1 \times ... \times M_k \text{ such that } m = (m_1, ..., r' m'_l + r'' m''_l, ..., m_k) \text{ for any } l \in J, \text{ any } r', r'' \in R, \text{ and any } m'_l, m''_l \in M_l (f (m) = r' f (m') + r'' f(m'') \text{ where } m' = (m_1, ..., m'_l, ..., m_k) \text{ and } m'' = (m_1, ..., m''_l, ..., m_k))\)
//


References


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