Topological properties of Taimanov semigroups

Oleg Gutik

Анотація


A semigroup $T$ is called a  Taimanov semigroup if $T$ contains two distinct elements $0,\infty$ such that $xy=\infty$ for any distinct points $x,y\in T\setminus\{0,\infty\}$ and $xy=0$ in all other cases.  We prove that any Taimanov semigroup $T$ has the following topological properties: (i) each $T_1$-topology with continuous shifts on $T$ is discrete; (ii) $T$ is closed in each $T_1$-topological semigroup containing $T$ as a subsemigroup; (iii) every non-isomorphic homomorphic image $Z$ of $T$ is a zero-semigroup and hence $Z$ is a topological semigroup in any topology on $Z$.

Ключові слова


Taimanov semigroup, semitopological semigroup, topological semigroup, discrete topology, zero-semigroup

Посилання


T.Banakh, I.Protasov, O.Sipacheva, Topologizations of a set endowed with an action of a monoid, Topology Appl. 169 (2014) 161--174.

S. Bardyla, Classifying locally compact semitopological polycyclic monoids, Math. Bull. Shevchenko Sci. Soc. 13 (2016) 31--38.

S.Bardyla, O.Gutik, On a semitopological polycyclic monoid, Algebra Discr. Math. 21:2 (2016) 163--183.

M.O. Bertman, T.T. West, Conditionally compact bicyclic semitopological semigroups, Proc. Roy. Irish Acad. A76:21--23 (1976) 219--226.

J.H. Carruth, J.A. Hildebrant, R.J. Koch,

The Theory of Topological Semigroups, Vols I and

II, Marcell Dekker, Inc., New York and Basel, 1983 and 1986.

I.Chuchman, O. Gutik,

Topological monoids of almost monotone injective co-finite partial selfmaps of the set of positive integers, Carpathian Math. Publ. 2:1 (2010) 119--132.

I. Chuchman, O. Gutik, On monoids of injective partial selfmaps almost everywhere the identity, Demonstr. Math. 44:4 (2011) 699--722.

A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups, Vols. I and II, Amer. Math. Soc. Surveys {bf 7}, Providence, R.I., 1961 and 1967.

C. Eberhart, J. Selden, On the closure of the bicyclic semigroup, Trans. Amer. Math. Soc. 144 (1969) 115--126.

R. Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.

I. Fihel, O. Gutik, On the closure of the extended bicyclic semigroup, Carpathian Math. Publ. 3:2 (2011) 131--157.

O. Gutik, On the dichotomy of a locally compact semitopological bicyclic monoid with adjoined zero,

Visn. L'viv. Univ., Ser. Mekh.-Mat. 80 (2015) 33--41.

O. Gutik, K. Maksymyk, On semitopological bicyclic extensions of linearly ordered groups, Mat. Metody Fiz.-Mekh. Polya (submitted) (arXiv:1608.00959).

O. Gutik, I. Pozdnyakova, On monoids of monotone injective partial selfmaps of $L_ntimes_{lex}mathbb{Z}$ with co-finite domains and images/i>, Algebra Discr. Math. 17:2 (2014) 256--279.

O. Gutik, D.Repovv{s}, Topological monoids of monotone, injective partial selfmaps of $mathbb{N}$ having cofinite domain and image,

Stud. Sci. Math. Hungar. 48:3 (2011) 342--353.

O. Gutik, D. Repovv{s}, On monoids of injective partial selfmaps of integers with cofinite domains and images,

Georgian Math. J. 19:3 (2012) 511--532.

A.A. Markov, On free topological groups, Izvestia Akad. Nauk SSSR 9 (1945), 3--64 (in Russian); English version in: Transl. Amer. Math. Soc. (1) 8M/b> (1962) 195--272.

Z. Mesyan, J. D. Mitchell, M. Morayne, Y. H. P'{e}resse,

emph{Topological graph inverse semigroups},

Topology Appl. 208 (2016) 106--126.

A.Yu. Ol'shanskiy, Remark on countable non-topologized groups, Vestnik Moscow Univ. Ser. Mech. Math.39 (1980) p.103 (in Russian).

W. Ruppert, Compact Semitopological Semigroups: An Intrinsic Theory, Lect. Notes Math., 1079, Springer, Berlin, 1984.

A.D. Taimanov, An example of a semigroup which admits only the discrete topology, Algebra i Logika 12:1 (1973) 114--116 (in Russian), English transl. in: Algebra and Logic 12:1 (1973), 64--65.

Y. Zelenyuk, On topologizing groups, J. Group Theory. 10:2 (2007) 235--244.


Повний текст: PDF (English)

Посилання

  • Поки немає зовнішніх посилань.


Creative Commons License
Ця робота ліцензована Creative Commons Attribution 3.0 License.