On characteristic properties of semigroups

Let K be a class of semigroups and P be a set of general properties of semigroups. We call a subset Q of P characteristic for a semigroup S ∈ K if, up to isomorphism and antiisomorphism, S is the only semigroup in K, which satisfies all the properties from Q. The set of properties P is called char...

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Date:	2015
Main Authors:	Bondarenko, V.M., Zaciha, Ya.V.
Format:	Article
Language:	English
Published:	Інститут прикладної математики і механіки НАН України 2015
Series:	Algebra and Discrete Mathematics
Online Access:	http://dspace.nbuv.gov.ua/handle/123456789/157998
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Cite this:	On characteristic properties of semigroups / V.M. Bondarenko, Ya.V. Zaciha // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 32–39. — Бібліогр.: 1 назв. — англ.

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spelling	irk-123456789-1579982019-06-23T01:24:50Z On characteristic properties of semigroups Bondarenko, V.M. Zaciha, Ya.V. Let K be a class of semigroups and P be a set of general properties of semigroups. We call a subset Q of P characteristic for a semigroup S ∈ K if, up to isomorphism and antiisomorphism, S is the only semigroup in K, which satisfies all the properties from Q. The set of properties P is called char-complete for K if for any S ∈ K the set of all properties P ∈ P, which hold for the semigroup S, is characteristic for S. We indicate a 7-element set of properties of semigroups which is a minimal char-complete set for the class of semigroups of order 3. 2015 Article On characteristic properties of semigroups / V.M. Bondarenko, Ya.V. Zaciha // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 32–39. — Бібліогр.: 1 назв. — англ. 1726-3255 2010 MSC:20M. http://dspace.nbuv.gov.ua/handle/123456789/157998 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Let K be a class of semigroups and P be a set of general properties of semigroups. We call a subset Q of P characteristic for a semigroup S ∈ K if, up to isomorphism and antiisomorphism, S is the only semigroup in K, which satisfies all the properties from Q. The set of properties P is called char-complete for K if for any S ∈ K the set of all properties P ∈ P, which hold for the semigroup S, is characteristic for S. We indicate a 7-element set of properties of semigroups which is a minimal char-complete set for the class of semigroups of order 3.
format	Article
author	Bondarenko, V.M. Zaciha, Ya.V.
spellingShingle	Bondarenko, V.M. Zaciha, Ya.V. On characteristic properties of semigroups Algebra and Discrete Mathematics
author_facet	Bondarenko, V.M. Zaciha, Ya.V.
author_sort	Bondarenko, V.M.
title	On characteristic properties of semigroups
title_short	On characteristic properties of semigroups
title_full	On characteristic properties of semigroups
title_fullStr	On characteristic properties of semigroups
title_full_unstemmed	On characteristic properties of semigroups
title_sort	on characteristic properties of semigroups
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/157998
citation_txt	On characteristic properties of semigroups / V.M. Bondarenko, Ya.V. Zaciha // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 32–39. — Бібліогр.: 1 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT bondarenkovm oncharacteristicpropertiesofsemigroups AT zacihayav oncharacteristicpropertiesofsemigroups
first_indexed	2025-07-14T10:26:10Z
last_indexed	2025-07-14T10:26:10Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 20 (2015). Number 1, pp. 32–39 © Journal “Algebra and Discrete Mathematics” On characteristic properties of semigroups Vitaliy M. Bondarenko, Yaroslav V. Zaciha Communicated by V. V. Kirichenko Abstract. Let K be a class of semigroups and P be a set of general properties of semigroups. We call a subset Q of P characteristic for a semigroup S ∈ K if, up to isomorphism and anti- isomorphism, S is the only semigroup in K, which satisfies all the properties from Q. The set of properties P is called char-complete for K if for any S ∈ K the set of all properties P ∈ P, which hold for the semigroup S, is characteristic for S. We indicate a 7-element set of properties of semigroups which is a minimal char-complete set for the class of semigroups of order 3. Introduction All properties of semigroups are assumed to be invariant with respect to isomorphism and anti-isomorphism. Let K be a class of semigroups and P be some set of general (qualitative and quantitative) properties of semigroups. For S ∈ K, we denote by P(S) the set of all properties P ∈ P which hold for the semigroup S. We say that a subset Q of P characteristic for a semigroup S ∈ K if, up to isomorphism and anti-isomorphism, S is the only semigroup in K, which satisfies all the properties from Q; if Q = {q1, . . . , qs}, then the properties q1, . . . , qs are called characteristic for S. Obviously, if Q ⊂ Q′ ⊆ P and Q is characteristic for S then so is Q′. The set of properties P is called char-complete for K if for any S ∈ K the subset P(S) of P is characteristic 2010 MSC: 20M. Key words and phrases: semigroup, anti-isomorphism, idempotent, Cayley table, characteristic property, char-complete set. V. M. Bondarenko, Y. V. Zaciha 33 for S. A char-complete set of properties is called minimal if it does not contain a proper char-complete subset of ones. In this paper we indicate a 7 properties of semigroups which form a minimal char-complete set for the class of semigroups of order 3. 1. Formulation of the main result We consider the following properties of a semigroup of order 3: P (C): commutativity; P (1): the existence of a unit element; P (0): the existence of a zero element; P +(0): the existence of an added zero element; Pid(1): the number of idempotents is equal to 1; Pid(2): the number of idempotents is equal to 2; Pgen(2): the smallest number of generators is equal to 2. The set of all these properties is denoted by P3(7). Our aim is to prove the following theorem. Theorem 1. The set P3(7) is a minimal char-complete set of properties for the class of semigroups of order 3. 2. Preliminaries In this section we present results from the paper [1]. Let S = {〈1〉, 〈2〉, . . . , 〈n〉} be a finite semigroup which is given by the Cayley table T . One wants to find some its minimal system of generators and the complete set of defining relations for these generators. In the first step one chooses an element 〈s〉 of S that is (according to the table) the product of two elements 〈i〉 6= 〈s〉 and 〈j〉 6= 〈s〉; then in the Cayley table T (including the header row and the header column) one substitutes 〈i〉〈j〉 instead of 〈s〉. The new table is denoted by T1. In the second step one chooses an element 〈s(1)〉 of the set S(1) = S\{〈s〉} that is the product of two elements i(1) and j(1), where i(1) = 〈i1〉, j(1) = 〈j1〉, or i(1) = 〈i1〉〈i2〉, j(1) = 〈j1〉, or i(1) = 〈i1〉, j(1) = 〈j1〉〈j2〉, or i(1) = 〈i1〉〈i2〉, j(1) = 〈j1〉〈j2〉, with 〈i1〉, 〈i2〉, 〈j1〉, 〈j2〉 6= 〈s1〉; then in the table T1 (including the header row and header column) one substitutes 〈i1〉〈j1〉 instead of 〈s1〉. The new table is denoted by T2. 34 On characteristic properties of semigroups In the next step one chooses an element s(2) of the set S(2) = S \ {〈s〉, 〈s(1)〉}, and so on. Upon completion of this process, say after m steps (m > 0), one has a minimal system of generators S(m) of the semigroup S (consisting of those elements of the header column of the last table that are of the form 〈k〉) and an appropriate set of defining relations in the form of the last table (which must be taken fully). Note that the specified process is ambiguous and so one can get different final system of generators. In the paper [1] this algorithm is applied to all semigroups of order 3 which are considered up to isomorphism and anti-isomorphism (if S be a semigroup, then a semigroup S′ with multiplication ◦ is called anti- isomorphic to S if S′ = S as sets and x ◦ y = yx). In each from 18 cases the algorithm has less than 3 steps. Under the transition from one table to another, the equality between the arrows specifies a replacement in the first table. 1) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈0〉〈1〉〈0〉〈0〉〈0〉〈2〉〈0〉〈0〉〈0〉 ⇒ (〈0〉 = 〈2〉2) ⇒ 〈2〉2〈2〉2〈2〉2 〈1〉〈2〉〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 〈1〉〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 〈2〉〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 2) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈0〉〈1〉〈0〉〈0〉〈0〉〈2〉〈0〉〈0〉〈1〉 ⇒ (〈1〉 = 〈2〉2) ⇒ 〈0〉〈2〉2〈2〉2〈2〉2 〈2〉〈0〉〈0〉〈0〉〈0〉〈2〉2〈2〉2〈2〉2 〈0〉〈0〉〈0〉〈2〉〈0〉〈0〉〈2〉2〈2〉2〈2〉2 ⇒ ⇒ (〈0〉 = 〈2〉2 · 〈2〉 = 〈2〉3) ⇒ 〈2〉3〈2〉3〈2〉3 〈2〉2 〈2〉〈2〉3〈2〉3〈2〉3 〈2〉3〈2〉3〈2〉3 〈2〉3〈2〉3〈2〉3 〈2〉3〈2〉3〈2〉3 〈2〉2 〈2〉3〈2〉3〈2〉3 〈2〉3〈2〉3〈2〉3 〈2〉3〈2〉3〈2〉3 〈2〉〈2〉3〈2〉3〈2〉3 〈2〉3〈2〉3〈2〉3 〈2〉2 V. M. Bondarenko, Y. V. Zaciha 35 3) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈0〉〈1〉〈0〉〈0〉〈0〉〈2〉〈0〉〈0〉〈2〉 ⇒ (〈0〉 = 〈1〉 · 〈2〉) ⇒ ⇒ 〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈2〉 4) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈0〉〈1〉〈0〉〈0〉〈0〉〈2〉〈0〉〈1〉〈2〉 ⇒ (〈0〉 = 〈1〉 · 〈2〉) ⇒ ⇒ 〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉〈2〉 5) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈0〉〈1〉〈0〉〈0〉〈0〉〈2〉〈2〉〈2〉〈2〉 ⇒ (〈0〉 = 〈1〉 · 〈2〉) ⇒ ⇒ 〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈2〉〈2〉〈2〉〈2〉 6) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈0〉〈1〉〈0〉〈0〉〈1〉〈2〉〈0〉〈1〉〈2〉 ⇒ (〈0〉 = 〈1〉2) ⇒ 〈1〉2〈1〉2〈1〉2 〈1〉〈2〉〈1〉2〈1〉2〈1〉2 〈1〉2〈1〉2〈1〉2 〈1〉2〈1〉2〈1〉2 〈1〉2〈1〉2〈1〉2 〈1〉〈1〉2〈1〉2〈1〉2 〈1〉2〈1〉2〈1〉2 〈1〉〈2〉〈1〉2〈1〉2〈1〉2 〈1〉〈2〉 36 On characteristic properties of semigroups 7) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈0〉〈1〉〈0〉〈1〉〈0〉〈2〉〈0〉〈0〉〈2〉 ⇒ (〈0〉 = 〈1〉 · 〈2〉) ⇒ ⇒ 〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈2〉 8) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈0〉〈1〉〈0〉〈1〉〈0〉〈2〉〈2〉〈2〉〈2〉 ⇒ (〈0〉 = 〈1〉 · 〈2〉) ⇒ ⇒ 〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈1〉 · 〈2〉〈2〉〈2〉〈2〉〈2〉 9) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈0〉〈1〉〈0〉〈1〉〈1〉〈2〉〈0〉〈1〉〈1〉 ⇒ (〈1〉 = 〈2〉2) ⇒ 〈0〉〈2〉2〈2〉2〈2〉2 〈2〉〈0〉〈0〉〈0〉〈0〉〈2〉2〈2〉2〈2〉2 〈0〉〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 〈2〉〈0〉〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 10) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈0〉〈1〉〈0〉〈1〉〈1〉〈2〉〈0〉〈1〉〈2〉 11) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈0〉〈1〉〈0〉〈1〉〈1〉〈2〉〈0〉〈2〉〈2〉 V. M. Bondarenko, Y. V. Zaciha 37 12) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈0〉〈1〉〈0〉〈1〉〈2〉〈2〉〈0〉〈2〉〈1〉 ⇒ (〈1〉 = 〈2〉2) ⇒ 〈0〉〈2〉2〈2〉2〈2〉2 〈2〉〈0〉〈0〉〈0〉〈0〉〈2〉2〈2〉2〈2〉2 〈0〉〈2〉2〈2〉2〈2〉2 〈2〉〈2〉〈0〉〈2〉〈2〉2〈2〉2〈2〉2 13) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈0〉〈1〉〈0〉〈1〉〈2〉〈2〉〈2〉〈2〉〈2〉 14) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈0〉〈1〉〈1〉〈1〉〈1〉〈2〉〈2〉〈2〉〈2〉 15) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈2〉〈1〉〈0〉〈0〉〈2〉〈2〉〈2〉〈2〉〈0〉 ⇒ (〈0〉 = 〈2〉2) ⇒ 〈2〉2〈2〉2〈2〉2 〈1〉〈2〉〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 〈2〉〈1〉〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 〈2〉〈2〉〈2〉〈2〉〈2〉2〈2〉2〈2〉2 16) 〈0〉〈1〉〈2〉〈0〉〈0〉〈0〉〈2〉〈1〉〈0〉〈1〉〈2〉〈2〉〈2〉〈2〉〈0〉 ⇒ (〈0〉 = 〈2〉2) ⇒ 〈2〉2〈2〉2〈2〉2 〈1〉〈2〉〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 〈2〉〈1〉〈2〉2〈2〉2〈2〉2 〈1〉〈2〉〈2〉〈2〉〈2〉〈2〉2〈2〉2〈2〉2 17) 〈0〉〈1〉〈2〉〈0〉〈0〉〈1〉〈1〉〈1〉〈1〉〈0〉〈0〉〈2〉〈1〉〈0〉〈0〉 ⇒ (〈1〉 = 〈0〉 · 〈2〉) ⇒ ⇒ 〈0〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈2〉〈0〉〈0〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈0〉〈0〉〈2〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈0〉 · 〈2〉〈0〉〈0〉 ⇒ 38 On characteristic properties of semigroups ⇒ (〈0〉 = 〈2〉2) ⇒ 〈2〉2〈2〉2〈2〉2 〈2〉3 〈2〉〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 〈2〉3 〈2〉3 〈2〉3 〈2〉3 〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 〈2〉〈2〉3 〈2〉2〈2〉2〈2〉2 〈2〉2〈2〉2〈2〉2 18) 〈0〉〈1〉〈2〉〈0〉〈0〉〈1〉〈2〉〈1〉〈1〉〈2〉〈0〉〈2〉〈2〉〈0〉〈1〉 ⇒ (〈2〉 = 〈1〉2) ⇒ 〈0〉〈1〉〈1〉2〈1〉2〈1〉2 〈0〉〈0〉〈1〉〈1〉2〈1〉2〈1〉2 〈1〉〈1〉〈1〉2〈1〉2〈1〉2 〈0〉〈1〉2〈1〉2〈1〉2 〈1〉2〈1〉2〈1〉2 〈0〉〈1〉 ⇒ ⇒ (〈0〉 = 〈1〉 · 〈1〉2 = 〈1〉3) ⇒ 〈1〉3〈1〉3〈1〉3 〈1〉〈1〉2 〈1〉3〈1〉3〈1〉3 〈1〉3〈1〉3〈1〉3 〈1〉〈1〉2 〈1〉〈1〉〈1〉2 〈1〉3〈1〉3〈1〉3 〈1〉2 〈1〉2 〈1〉3〈1〉3〈1〉3 〈1〉 From the above, it easily follows the next statement (which was not formulated in [1]) . Theorem 2. Let S be a semigroup of order 3. Then any two its minimal systems of generators are of the same order, and coincide if S is not a group. Note that the group of order 3 is given by the case 18). 3. Proof of Theorem 1 The following table T , which follows from the results of section 2, shows what the properties hold for the semigroups 1) – 18) (“+” means that the corresponding property holds, and its absence means that the corresponding property does not hold). Since all rows of this table (without the header row and the header co- lumn) are mutually different, the set of properties P3(7) is char-complete. To prove that the char-complete set P3(7) is minimal it suffices to verify that the table T without any fix column X (and, of course, without the header row and the header column) has two equal rows. It is easy to see that if X is equal to C, P (1), P (0), P +(0), Pid(1), Pid(2), Pgen(2), then, respectively, the following two rows are equal: 3 and 4, 13 and 14, 4 and 5, 3 and 9, 1 and 7, 5 and 8, 8 and 14. V. M. Bondarenko, Y. V. Zaciha 39 Table 1. T C P (1) P (0) P +(0) Pid(1) Pid(2) Pgen(2) 1 + + + + 2 + + + 3 + + + + 4 + + + 5 + + 6 + + + + + 7 + + + 8 + 9 + + + + + 10 + + + + 11 + + 12 + + + + + + 13 + 14 15 + + + 16 + + + + 17 + + 18 + + + References [1] V. M. Bondarenko, Y. V. Zaciha, On defining relations for minimal generator systems of three-order semigroups, Science Journal of National Pedagogical Drago- manov University, Series 1: Physics and Mathematics (2013), no. 14, 62-67 (in Ukrainian). Contact information V. M. Bondarenko, Y. V. Zaciha Institute of Mathematics, Tereshchenkivska 3, 01601 Kyiv, Ukraine E-Mail(s): vit-bond@imath.kiev.ua, zaciha@mail.ru Received by the editors: 07.09.2015 and in final form 07.09.2015.

On characteristic properties of semigroups

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