Difference between revisions of "Gauss semi-group"
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| − | A commutative semi-group with unit satisfying the cancellation law, in which any non-invertible element $a$ is decomposable into a product of irreducible elements (i.e. elements that cannot be represented as a product of non-invertible factors); moreover, for each two such decompositions | + | A commutative [[semi-group]] with unit (i.e. [[Monoid]]) satisfying the cancellation law, in which any non-invertible element $a$ is decomposable into a product of irreducible elements (i.e. non-invertible elements that cannot be represented as a non-trivial product of non-invertible factors); moreover, for each two such decompositions |
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a = b_1 \cdots b_k\ \ \text{and}\ \ a = c_1 \cdots c_l | a = b_1 \cdots b_k\ \ \text{and}\ \ a = c_1 \cdots c_l | ||
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| − | where $\epsilon_1,\ldots,\epsilon_k$ are invertible elements. Typical examples of Gauss semi-groups include the multiplicative semi-group of non-zero integers, and that of non-zero polynomials in one unknown over a field. Any two elements of a Gauss semi-group have a highest common divisor. | + | where $\epsilon_1,\ldots,\epsilon_k$ are invertible elements. |
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| + | Typical examples of Gauss semi-groups include the multiplicative semi-group of non-zero integers, and that of non-zero polynomials in one unknown over a field. Any two elements of a Gauss semi-group have a highest common divisor. | ||
====References==== | ====References==== | ||
Revision as of 19:02, 12 October 2014
A commutative semi-group with unit (i.e. Monoid) satisfying the cancellation law, in which any non-invertible element $a$ is decomposable into a product of irreducible elements (i.e. non-invertible elements that cannot be represented as a non-trivial product of non-invertible factors); moreover, for each two such decompositions $$ a = b_1 \cdots b_k\ \ \text{and}\ \ a = c_1 \cdots c_l $$ one has $k=l$ and, possibly after renumbering the factors, also $$ b_1 = c_1 \epsilon_1,\ \ldots,\ b_k = c_k \epsilon_k $$
where $\epsilon_1,\ldots,\epsilon_k$ are invertible elements.
Typical examples of Gauss semi-groups include the multiplicative semi-group of non-zero integers, and that of non-zero polynomials in one unknown over a field. Any two elements of a Gauss semi-group have a highest common divisor.
References
| [1] | A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) |
Gauss semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_semi-group&oldid=33586