×

zbMATH — the first resource for mathematics

Green index and finiteness conditions for semigroups. (English) Zbl 1172.20045
Given a semigroup \(S\) and a subsemigroup \(T\), the Rees index of \(T\) in \(S\) is defined to be the cardinality of the complement \(S\setminus T\). In this article, in Section 2, the authors give basic properties of Green index. Then in Section 3 and Section 4, they prove the following main results of their paper:
Let \(S\) be a semigroup and let \(T\) be a subsemigroup of \(S\) with finite Green index. Let \(\Gamma_i\) (\(i\in I\)) be the Schützenberger groups of the \(T\)-relative \(H\)-classes of the complement \(S\setminus T\). Then the following hold: (I) \(S\) is locally finite if and only if \(T\) is locally finite, in which case every group \(\Gamma_i\) is locally finite; (II) \(S\) is periodic if and only if \(T\) is periodic, in which case every group \(\Gamma_i\) is periodic; (III) \(S\) has finitely many right ideals if and only if \(T\) has finitely many right ideals (and the dual result for left ideals); (IV) \(S\) is residually finite if and only if \(T\) and \(\Gamma_i \) are all residually finite.
In Section 5, the authors define the relationship between Green index and syntactic index. Finally, they give some applications of their results.

MSC:
20M10 General structure theory for semigroups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anscombre, J.-C., Sur une extension du lemme de Green, Atti accad. naz. lincei rend. cl. sci. fis. mat. natur. (8), 55, 650-656, (1974), 1973 · Zbl 0305.20037
[2] C. Carvalho, R. Gray, N. Ruškuc, Kernels of finite index in inverse semigroups, submitted for publication
[3] Clark, C.E.; Carruth, J.H., Generalized Green’s theories, Semigroup forum, 20, 2, 95-127, (1980) · Zbl 0438.20045
[4] Eilenberg, S., Automata, languages, and machines, vol. A, Pure appl. math., vol. 58, (1974), Academic Press [A subsidiary of Harcourt Brace Jovanovich Publishers] New York · Zbl 0317.94045
[5] Evans, T., Some connections between residual finiteness, finite embeddability and the word problem, J. London math. soc. (2), 1, 399-403, (1969) · Zbl 0184.03502
[6] R. Gray, N. Ruškuc, Generators and relations for subsemigroups via boundaries in Cayley graphs, submitted for publication
[7] Green, J.A., On the structure of semigroups, Ann. of math. (2), 54, 163-172, (1951) · Zbl 0043.25601
[8] Hoffmann, M.; Thomas, R.M.; Ruškuc, N., Automatic semigroups with subsemigroups of finite Rees index, Internat. J. algebra comput., 12, 3, 463-476, (2002) · Zbl 1010.20040
[9] Howie, J.M., Fundamentals of semigroup theory, London math. soc. monogr., vol. 7, (1995), Academic Press [Harcourt Brace Jovanovich Publishers] London · Zbl 0835.20077
[10] Inata, I.; Kobayashi, Y., Monoids with subgroups of finite index and the braid inverse monoid, Japanese association of mathematical sciences 2001 annual meeting, Tennoji, Sci. math. jpn., 57, 1, 3-9, (2003) · Zbl 1048.20036
[11] Jura, A., Coset enumeration in a finitely presented semigroup, Canad. math. bull., 21, 1, 37-46, (1978) · Zbl 0379.20047
[12] Jura, A., Determining ideals of a given finite index in a finitely presented semigroup, Demonstratio math., 11, 3, 813-827, (1978) · Zbl 0412.20066
[13] Jura, A., Some remarks on nonexistence of an algorithm for finding all ideals of a given finite index in a finitely presented semigroup, Demonstratio math., 13, 2, 573-578, (1980) · Zbl 0465.20056
[14] Lallement, G., Semigroups and combinatorial applications, Pure appl. math., (1979), John Wiley & Sons New York/Chichester/Brisbane · Zbl 0421.20025
[15] Lawson, M.V., Inverse semigroups, () · Zbl 0820.20070
[16] Linton, S.A.; Pfeiffer, G.; Robertson, E.F.; Ruškuc, N., Groups and actions in transformation semigroups, Math. Z., 228, 3, 435-450, (1998) · Zbl 0902.20028
[17] Malheiro, A., On trivializers and subsemigroups, (), 188-204 · Zbl 1127.20035
[18] Márki, L.; Steinfeld, O., A generalization of Green’s relations in semigroups, Collection of articles dedicated to Alfred hoblitzelle Clifford on the occasion of his 65th birthday and to Alexander doniphan Wallace on the occasion of his 68th birthday, Semigroup forum, 7, 1-4, 74-85, (1974) · Zbl 0276.20052
[19] Nikolov, N., On subgroups of finite index in positively finitely generated groups, Bull. London math. soc., 37, 6, 873-877, (2005) · Zbl 1097.20028
[20] Pastijn, F., A representation of a semigroup by a semigroup of matrices over a group with zero, Semigroup forum, 10, 3, 238-249, (1975) · Zbl 0299.20057
[21] Pride, S.J.; Wang, J., Subgroups of finite index in groups with finite complete rewriting systems, Proc. edinb. math. soc. (2), 43, 1, 177-183, (2000) · Zbl 0976.20025
[22] Ruškuc, N., On large subsemigroups and finiteness conditions of semigroups, Proc. London math. soc. (3), 76, 2, 383-405, (1998) · Zbl 0891.20036
[23] Ruškuc, N., On finite presentability of monoids and their schützenberger groups, Pacific J. math., 195, 2, 487-509, (2000) · Zbl 1009.20065
[24] Ruškuc, N.; Thomas, R.M., Syntactic and Rees indices of subsemigroups, J. algebra, 205, 2, 435-450, (1998) · Zbl 0914.20053
[25] Smith, M.G.; Wilson, J.S., On subgroups of finite index in compact Hausdorff groups, Arch. math. (basel), 80, 2, 123-129, (2003) · Zbl 1039.20011
[26] Wallace, A.D., Relative ideals in semigroups. II. the relations of Green, Acta math. acad. sci. hungar, 14, 137-148, (1963) · Zbl 0122.26802
[27] Wang, J., Finite complete rewriting systems and finite derivation type for small extensions of monoids, J. algebra, 204, 2, 493-503, (1998) · Zbl 0944.20040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.