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''of a semi-group''
 
''of a semi-group''
  
A Rees quotient semi-group (cf. [[Semi-group|Semi-group]]) of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746801.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746802.png" /> is the two-sided [[Principal ideal|principal ideal]] of the semi-group generated by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746803.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746804.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746805.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746806.png" />-class (cf. [[Green equivalence relations|Green equivalence relations]]) containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746807.png" />. If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746808.png" /> is not empty, then it is an ideal, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p0746809.png" />, one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468010.png" />. A principal factor of a semi-group is also known as an ideal factor. An arbitrary principal factor of a semi-group is either a semi-group with zero multiplication, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468011.png" />-simple semi-group or an ideally-simple semi-group (cf. [[Simple semi-group|Simple semi-group]]). The last situation occurs if and only if the semi-group has a kernel (cf. [[Kernel of a semi-group|Kernel of a semi-group]]) and this kernel coincides with the given principal factor. A semi-group without a principal factor with zero multiplication is said to be semi-simple; the condition of semi-simplicity of a semi-group is equivalent, for example, to the fact that for any of its two-sided ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468012.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468013.png" /> is valid. All regular semi-groups (cf. [[Regular semi-group|Regular semi-group]]) are semi-simple. If each principal factor of a semi-group is either completely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468014.png" />-simple or completely simple (cf. [[Completely-simple semi-group|Completely-simple semi-group]]), then the semi-group is called completely semi-simple. A semi-group is completely semi-simple if and only if it is regular and satisfies any of the following (mutually-dual) conditions: for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468015.png" />-class the partially ordered set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468016.png" />-classes (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468017.png" />-classes) contained in it has a minimal element; in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468018.png" />.
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A Rees quotient semi-group (cf. [[Semi-group|Semi-group]]) of the form $J(x)/N(x)$, where $J(x)$ is the two-sided [[Principal ideal|principal ideal]] of the semi-group generated by the element $x$ and $N(x)=J(x)\setminus J_x$, where $J_x$ is the $\mathcal J$-class (cf. [[Green equivalence relations|Green equivalence relations]]) containing $x$. If the set $N(x)$ is not empty, then it is an ideal, and if $N(x)=\emptyset$, one puts $J(x)/N(x)=J(x)$. A principal factor of a semi-group is also known as an ideal factor. An arbitrary principal factor of a semi-group is either a semi-group with zero multiplication, a $0$-simple semi-group or an ideally-simple semi-group (cf. [[Simple semi-group|Simple semi-group]]). The last situation occurs if and only if the semi-group has a kernel (cf. [[Kernel of a semi-group|Kernel of a semi-group]]) and this kernel coincides with the given principal factor. A semi-group without a principal factor with zero multiplication is said to be semi-simple; the condition of semi-simplicity of a semi-group is equivalent, for example, to the fact that for any of its two-sided ideals $A$ the equality $A^2=A$ is valid. All regular semi-groups (cf. [[Regular semi-group|Regular semi-group]]) are semi-simple. If each principal factor of a semi-group is either completely $0$-simple or completely simple (cf. [[Completely-simple semi-group|Completely-simple semi-group]]), then the semi-group is called completely semi-simple. A semi-group is completely semi-simple if and only if it is regular and satisfies any of the following (mutually-dual) conditions: for each $\mathcal J$-class the partially ordered set of $\mathcal L$-classes (or $\mathcal R$-classes) contained in it has a minimal element; in this case $\mathcal J=\mathcal D$.
  
Any semi-group consists, as it were, of its principal factors. This explains, in particular, the important role played by ideally-simple and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074680/p07468019.png" />-simple semi-groups in the theory of semi-groups.
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Any semi-group consists, as it were, of its principal factors. This explains, in particular, the important role played by ideally-simple and $0$-simple semi-groups in the theory of semi-groups.
  
 
====References====
 
====References====

Revision as of 15:22, 15 April 2014

of a semi-group

A Rees quotient semi-group (cf. Semi-group) of the form $J(x)/N(x)$, where $J(x)$ is the two-sided principal ideal of the semi-group generated by the element $x$ and $N(x)=J(x)\setminus J_x$, where $J_x$ is the $\mathcal J$-class (cf. Green equivalence relations) containing $x$. If the set $N(x)$ is not empty, then it is an ideal, and if $N(x)=\emptyset$, one puts $J(x)/N(x)=J(x)$. A principal factor of a semi-group is also known as an ideal factor. An arbitrary principal factor of a semi-group is either a semi-group with zero multiplication, a $0$-simple semi-group or an ideally-simple semi-group (cf. Simple semi-group). The last situation occurs if and only if the semi-group has a kernel (cf. Kernel of a semi-group) and this kernel coincides with the given principal factor. A semi-group without a principal factor with zero multiplication is said to be semi-simple; the condition of semi-simplicity of a semi-group is equivalent, for example, to the fact that for any of its two-sided ideals $A$ the equality $A^2=A$ is valid. All regular semi-groups (cf. Regular semi-group) are semi-simple. If each principal factor of a semi-group is either completely $0$-simple or completely simple (cf. Completely-simple semi-group), then the semi-group is called completely semi-simple. A semi-group is completely semi-simple if and only if it is regular and satisfies any of the following (mutually-dual) conditions: for each $\mathcal J$-class the partially ordered set of $\mathcal L$-classes (or $\mathcal R$-classes) contained in it has a minimal element; in this case $\mathcal J=\mathcal D$.

Any semi-group consists, as it were, of its principal factors. This explains, in particular, the important role played by ideally-simple and $0$-simple semi-groups in the theory of semi-groups.

References

[1] E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
[2] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)


Comments

A principal factor of a semi-group is also called a chief factor of a semi-group.

How to Cite This Entry:
Principal factor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_factor&oldid=19069
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article