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Homesubcommutative

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# subcommutative

A semigroup $(S,\,\cdot)$ is said to be *left subcommutative* if for any two of its elements $a$ and $b$, there exists its element $c$ such that

$\displaystyle ab=ca.$ | (1) |

A semigroup $(S,\,\cdot)$ is said to be *right subcommutative* if for any two of its elements $a$ and $b$, there exists its element $d$ such that

$\displaystyle ab=bd.$ | (2) |

If $S$ is both left subcommutative and right subcommutative, it is *subcommutative*.

The commutativity is a special case of all the three kinds of subcommutativity.

Example 1. The following operation table defines a right subcommutative semigroup $\{0,\,1,\,2,\,3\}$ which is not left subcommutative (e.g. $0\!\cdot\!3=2\neq c\!\cdot\!0$):

$\begin{array}[]{c|cccc}\cdot&0&1&2&3\\ \hline\;0&0&0&2&2\\ \;1&0&1&2&3\\ \;2&0&0&2&2\\ \;3&0&1&2&3\end{array}$ |

Example 2. The multiplicative group of the square matrices over a field is both left and right subcommutative (but not commutative), since the equations (1) and (2) are satisfied by

$c\;=\;aba^{{-1}}\quad\mbox{and}\quad d\;=\;b^{{-1}}ab.$ |

Remark. One uses the above attributes also for a ring $(S,\,+,\,\cdot)$ if its multiplicative semigroup $(S,\,\cdot)$ satisfies the corresponding requirements.

# References

- 1
S. Lajos: “On $(m,\,n)$-ideals in subcommutative semigroups”. –
*Elemente der Mathematik*24 (1969). - 2
V. P. Elizarov: “Subcommutative Q-rings”. –
*Mathematical notes*2 (1967).

## Mathematics Subject Classification

20M25*no label found*20M99

*no label found*

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