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where, for the sake of simplicity, the same symbol
, which aggregates elements in
multiset polynomial notation, here denotes the multiset sum operation.
Multiset sum is, of course, commutative and associative, that is for any multisets
+
,
,
X
Y
Z :
X
+
Y
=
Y
+
X
(
X
+
Y
)+
Z
=
X
+(
Y
+
Z
) .
The multiset without elements is denoted by 0 (which also denotes the empty set).
Multiset multiplication m
·
X of a multiset X ,overaset A , by a natural number
m is defined by the following equation:
(
m
·
X
)(
a
)=
m
·
X
(
a
)
where symbol
on the right hand denotes the product between numbers. If we
identify an element a as a multiset of a single object (with multiplicity 1), then
m
·
·
a
=
ma , that is, for single multisets, multiplication can be identified with multi-
plicity.
Multiset operations naturally occur in chemistry. In fact, a chemical reaction is
an operation transforming a multiset of molecules (reactants) into another multiset
of molecules (products).
A chemical reaction is usually denoted by an ordered pair of two multisets, with
an arrow between them:
Reactants
Products
A reaction such as:
X
+
Y
2 Z
can be viewed as an operation which takes the molecules, viewed as multisets of
atoms, X and Y and transforms them into the multiset 2 Z . The elements on the left
of the arrow are also called substrates of the reaction.
Let us consider a reaction of n reactants X 1 ,
Y m :
The so called stoichiometric balance of such a reaction is the procedure which
provides the minimum multiplicities (if they exist) h 1 ,
X 2 ,...
X n and m products Y 1 ,
Y 2 ,...
h 2 ,...
h n ,
k 1 ,
k 2 ,...
k m of reac-
tants and products such that:
(
h 1 ·
X 1 )+(
h 2 ·
X 2 )+ ... (
h n ·
X n )=(
k 1 ·
Y 1 )+(
k 2 ·
Y 2 )+ ... (
k m ·
Y m )
The law of multiple proportions is one of the fundamental laws of stoichiometry
and was first discovered by the English chemist John Dalton in 1803. The law states
that when chemical elements combine, they do so in a ratio of whole numbers:
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