View source for Semi-group with a finiteness condition

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A [[Semi-group|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|Periodic semi-group]]), local finiteness (see [[Locally finite semi-group|Locally finite semi-group]]), residual finiteness (see [[Residually-finite semi-group|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 [[#References|[8]]].

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 [[#References|[2]]]. 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 [[#References|[3]]], [[#References|[4]]]).

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|Ideal series]] of a semi-group) each factor of which is a [[Brandt semi-group|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 [[#References|[5]]]).

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|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|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 [[#References|[6]]].

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 [[#References|[6]]]. 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 [[#References|[7]]] (is finitely generated).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,   G.B. Preston,   "Algebraic theory of semi-groups" , '''2''' , Amer. Math. Soc.  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.N. Shevrin,   "Some finiteness conditions in semigroup theory"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''29''' :  3  (1965)  pp. 553–566  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.N. Shevrin,   "A general theorem concerning semi-groups with certain finiteness conditions"  ''Math. Notes'' , '''15''' :  6  (1974)  pp. 552–557  ''Mat. Zametki'' , '''15''' :  6  (1974)  pp. 925–935</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.N. Shevrin,   "On the theory of periodic semigroups"  ''Izv. Vuzov. Mat.'' , '''5'''  (1974)  pp. 205–215  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  T.I. Ershova,   "Inverse semigroups with certain finiteness conditions"  ''Izv. Vyzov. Mat.'' , '''11'''  (1977)  pp. 7–14  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E. Hotzel,   "On finiteness conditions in semigroups"  ''J. of Algebra'' , '''60''' :  2  (1979)  pp. 352–370</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  I.B. Kozhukov,   "On semigroups with minimal or maximal condition on left congruences"  ''Semigroup Forum'' , '''21''' :  4  (1980)  pp. 337–350</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  F. Pastijn,   "Embedding semigroups in semibands"  ''Semigroup Forum'' , '''14''' :  3  (1977)  pp. 247–264</TD></TR></table>