Semi-group with a finiteness condition
A semi-group possessing a property $\theta$ which is valid for all finite semi-groups (a property of this kind is called a finiteness condition). The definition of the property $\theta$ may be phrased in terms of the elements of the semi-group, its sub-semi-groups, etc.
Examples of finiteness conditions are: periodicity (see Periodic semi-group), local finiteness (see Locally finite semi-group), residual finiteness (see Residually-finite semi-group), finite generation, and finite presentation. The investigation of finitely-presented semi-groups belongs largely to the field of algorithmic problems. The most well-known condition under which a finitely-generated semi-group is also finitely presented is commutativity (Redei's theorem). Any countable semi-group can be imbedded in a semi-group with two generators, and also in a semi-group with three idempotent generators .
A broad range of finiteness conditions is phrased in terms of the lattice of sub-semi-groups (e.g. the minimum condition for sub-semi-groups). A semi-group $S$ satisfies the minimum condition for sub-semi-groups if and only if it is periodic, has only finitely many torsion classes, if in each torsion class $K_e$ the maximal subgroup $G_e$ satisfies the minimum condition for sub-semi-groups, and if the difference $K_e/G_e$ is finite . Semi-groups of finite rank have a similar structure (finite rank means that the minimum number of generators of each finitely-generated sub-semi-group of $S$ does not exceed a fixed number); the same is true of semi-groups of finite breadth (i.e. any finite set $M$ of elements of $S$ contains a subset generating the same sub-semi-group as $M$, and the number of elements of which does not exceed a fixed number); of periodic semi-groups with the maximum condition for sub-semi-groups; etc. (see , ).
An inverse semi-group satisfies the minimum condition for inverse sub-semi-groups if and only if it has a principal series (see Ideal series of a semi-group) each factor of which is a Brandt semi-group with finitely many idempotents, all maximal subgroups of which satisfy the minimum condition for subgroups. Analogous descriptions have been obtained for the maximum condition, finiteness of rank, etc. (see ).
Some finiteness conditions are formulated in terms of the partially ordered set of ideals of the semi-group. The best known of these are the minimum conditions $M_L$, $M_R$, $M_J$ for principal left, right and two-sided ideals, respectively (these conditions are often defined in terms of $\mathcal L$-, $\mathcal R$- and $\mathcal J$-classes; see Green equivalence relations). The definition of the condition $M_H$ for $\mathcal H$-classes is similar. The conjunction of the conditions $M_L$ and $M_R$ is equivalent to the conjunction of the conditions $M_J$ and $M_H$, but otherwise these conditions are independent; in particular, a semi-group with the conditions $M_L$ and $M_J$ does not necessarily satisfy the conditions $M_R$ and $M_H$. At the same time, a semi-simple (see Principal factor of a semi-group) semi-group with the condition $M_L$ or $M_R$ satisfies $M_J$. For regular semi-groups all four conditions are equivalent; any semi-group with the condition $M_H$ is quasi-periodic. A finitely-generated semi-group with the condition $M_L$ or $M_R$ and all subgroups of which are finite, is itself finite .
A semi-group with the minimum condition for right congruences is periodic, satisfies condition $M_L$ and the dual maximum condition for principal left ideals; if at the same time all its subgroups are finite, then the semi-group itself is finite . In inverse semi-groups, the minimum condition for left congruences, and also the condition that the semi-group have only finitely many idempotents and satisfy the minimum condition for one-sided congruences have been studied. A commutative semi-group satisfies the minimum (maximum) condition for congruences if and only if it has a principal series and satisfies the minimum condition for subgroups  (is finitely generated).
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Semi-group with a finiteness condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-group_with_a_finiteness_condition&oldid=32654