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\(\aleph_0\)-categoricity of semigroups. II. (English) Zbl 1495.20048

Summary: A countable semigroup is \(\aleph_0\)-categorical if it can be characterised, up to isomorphism, by its first-order properties. In this paper we continue our investigation into the \(\aleph_0\)-categoricity of semigroups. Our main results are a complete classification of \(\aleph_0\)-categorical orthodox completely \(0\)-simple semigroups, and descriptions of the \(\aleph_0\)-categorical members of certain classes of strong semilattices of semigroups.
For Part I see [V. Gould and the author, ibid. 99, No. 2, 260–292 (2019; Zbl 1467.20061)].

MSC:

20M10 General structure theory for semigroups
03C60 Model-theoretic algebra
03C35 Categoricity and completeness of theories
20M19 Orthodox semigroups

Citations:

Zbl 1467.20061
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References:

[1] Apps, AB, On the structure of \(\aleph_0\)-categorical groups, J. Algebra, 81, 320-339 (1982) · Zbl 0512.20013
[2] Araújo, J.; Bünau, PV; Mitchell, JD; Neunhöffer, M., Computing automorphisms of semigroups, J. Symb. Comput., 45, 373-392 (2010) · Zbl 1186.20049
[3] Engeler, E., A characterization of theories with isomorphic denumerable models, Am. Math. Soc. Not., 6, 161 (1959)
[4] Erdős, P.; Spencer, J., Probabilistic Methods in Combinatorics (1974), New York: Academic Press, New York · Zbl 0308.05001
[5] Evans, DM, Model Theory of Groups and Automorphism Groups (1997), Cambridge: Cambridge University Press, Cambridge · Zbl 0878.03026
[6] Droste, M., Structure of partially ordered sets with transitive automorphism groups, Mem. Am. Math. Soc., 57, 334 (1985) · Zbl 0574.06001
[7] Droste, M.; Kuske, D.; Truss, JK, On homogeneous semilattices and their automorphism groups, Order, 16, 31-56 (1999) · Zbl 0945.06001
[8] Goldstern, M.; Grossberg, R.; Kojman, M., Infinite homogeneous bipartite graphs with unequal sides, Discret. Math., 149, 69-82 (1996) · Zbl 0843.05050
[9] Gould, V.; Quinn-Gregson, T., \( \aleph_0\)-categoricity of semigroups, Semigroup Forum, 99, 260-292 (2019) · Zbl 1467.20061
[10] Graham, RL, On finite 0-simple semigroups and graph theory, Math. Syst. Theory, 2, 325-339 (1968) · Zbl 0177.03103
[11] Graham, N.; Graham, R.; Rhodes, J., Maximal subsemigroups of finite semigroups, J. Comb. Theory, 4, 203-209 (1968) · Zbl 0157.04901
[12] Grzegorczyk, A., Logical uniformity by decomposition and categoricity in \(\aleph_0\), Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 16, 687-692 (1968) · Zbl 0184.01204
[13] Hall, TE, On regular semigroups whose idempotents form a subsemigroup, Bull. Aust. Math. Soc., 1, 195-208 (1969) · Zbl 0172.31101
[14] Hodges, W., Model Theory (1993), Cambridge: Cambridge University Press, Cambridge · Zbl 0789.03031
[15] Houghton, CH, Completely 0-simple semigroups and their associated graphs and groups, Semigroup Forum, 14, 41-67 (1977) · Zbl 0358.20071
[16] Howie, JM, Idempotents in completely 0-simple semigroups, Glasgow Math. J., 19, 109-113 (1978) · Zbl 0393.20044
[17] Howie, JM, Fundamentals of Semigroup Theory (1995), Oxford: Oxford University Press, Oxford · Zbl 0835.20077
[18] Jackson, M.; Volkov, M., Undecidable problems for completely 0-simple semigroups, J. Pure Appl. Algebra, 213, 1961-1978 (2009) · Zbl 1179.20052
[19] Macpherson, HD, A survey of homogeneous structures, Discret. Math., 311, 1599-1634 (2011) · Zbl 1238.03032
[20] Petrich, M.; Reilly, NR, Completely Regular Semigroups (1999), New York: Wiley, New York · Zbl 0967.20034
[21] Quinn-Gregson, T.: Homogeneity and \(\aleph_0\)-categoricity of semigroups. Ph.D thesis, University of York (2017)
[22] Quinn-Gregson, T., Homogeneous bands, Adv. Math., 328, 623-660 (2018) · Zbl 1494.20086
[23] Quinn-Gregson, T., Homogeneity of inverse semigroups, Int. J. Algebr. Comput., 28, 837-875 (2018) · Zbl 1491.20124
[24] Quinn-Gregson, T., Homogeneous completely simple semigroups, Mathematika, 66, 733-751 (2020) · Zbl 1508.20088
[25] Rees, D., On semi-groups, Proc. Camb. Phil. Soc., 36, 387-400 (1940) · Zbl 0028.00401
[26] Rhodes, J., Steinberg, B.: The \(q\)-Theory of Finite Semigroups. Springer Monographs in Maths. Springer, Berlin (2009) · Zbl 1186.20043
[27] Rosenstein, JG, \( \aleph_0\)-categoricity of linear orders, Fund. Math., 64, 1-5 (1969) · Zbl 0179.01303
[28] Ryll-Nardzewski, C., On the categoricity in power \(\le \aleph_0\), Bull. Acad. Polon. Ser. Sci. Math. Astro. Phys., 7, 545-548 (1959) · Zbl 0117.01101
[29] Svenonius, L., \( \aleph_0\)-categoricity in first-order predicate calculus, Theoria, 25, 82-94 (1959)
[30] Worawiset, S.: The structure of endomorphism monoids of strong semilattices of left simple semigroups. Ph.D thesis, University of Oldenburg (2011)
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