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 [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 [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 [3], [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 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 [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). 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 [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 [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 [7] (is finitely generated).

References

[1]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 2 , Amer. Math. Soc. (1967)
[2]	L.N. Shevrin, "Some finiteness conditions in semigroup theory" Izv. Akad. Nauk SSSR Ser. Mat. , 29 : 3 (1965) pp. 553–566 (In Russian)
[3]	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
[4]	L.N. Shevrin, "On the theory of periodic semigroups" Izv. Vuzov. Mat. , 5 (1974) pp. 205–215 (In Russian)
[5]	T.I. Ershova, "Inverse semigroups with certain finiteness conditions" Izv. Vyzov. Mat. , 11 (1977) pp. 7–14 (In Russian)
[6]	E. Hotzel, "On finiteness conditions in semigroups" J. of Algebra , 60 : 2 (1979) pp. 352–370
[7]	I.B. Kozhukov, "On semigroups with minimal or maximal condition on left congruences" Semigroup Forum , 21 : 4 (1980) pp. 337–350
[8]	F. Pastijn, "Embedding semigroups in semibands" Semigroup Forum , 14 : 3 (1977) pp. 247–264