Multiensemble

See also: Bag (homonymy)

A multiensemble (sometimes called bag ) is a pair $(has,\; m)$ where $A$ is an unspecified unit called support and $m$ a function of $A$ in the whole of the natural Entiers, called multiplicity.

A multiensemble is known as finished if the sum of the multiplicities of the elements of its support is finished.

Intuitively, such an object can be seen as a Ensemble of elements of $A$ where an element can appear several times, in fact an element $x$ will appear $m\; (X)$ time. This justifies the abstract notation of the finished multiensembles: $\backslash \{\backslash !\backslash !\; \backslash \; \{has,\; B,\; has,\; B,\; B,\; D\; \backslash \}\; \backslash !\; \backslash !\; \backslash \}$ represents the multiensemble $(\backslash \; \{has,\; B,\; C,\; D\; \backslash \},\; m)$ where $m$ is the function of such as $m\; (a)=2$, $m\; (b)=3$, $m\; (c)=0$ and $m\; (d)=1$.

One can also see it like a commutative list , i.e. which one can permute the elements, in other words a commutative Monoïde free.

Order multiensemble

If one provided $A$ with an order >, it is possible to define an order between the multiensembles of support $A$ which one order multiensemble calls: $(has,\; m)$ is strictly larger than $(has,\; N)$ for the order multiensemble if

• $m\; \backslash \; neq\; n$ and
• for all $a\; \backslash \; in\; A$, if $\backslash ,\; N\; (a)\; >\; m\; (a)$ then it exists $a\text{'}\; \backslash \; in\; A$ such as $\backslash ,\; a\text{'}\; >\; a$ and $\backslash ,\; m\; (a\text{'})\; >\; N\; (a\text{'})$.

Intuitively, in other words, an unspecified number of elements of $(has,\; N)$ can be replaced by a larger element to obtain $(has,\; m)$.

Example: if one orders the letters in the alphabetical order (

If it is supposed that the order on $A$ is quite founded, then the order multiensemble thus defined is too. This property is sometimes called “Hydre de Lerne”: it is supposed that when Hercules cuts a head, an unspecified number (finished) of heads can push back, but they all are strictly smaller. Then one is sure that Hercules will come to end from the hydre.

Simple: Multiset

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