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УДК 512.53
DOI 10.25513/1812-3996.2018.23(3).56-58
О ПОДПОЛУГРУППАХ СВОБОДНЫХ ЛЕВОРЕГУЛЯРНЫХ ПОЛУГРУПП (посвящается памяти профессора Георгия Петровича Кукина)
А. Н. Шевляков
Институт математики им. С. Л. Соболева СО РАН, Омский филиал, г. Омск, Россия
Информация о статье Аннотация. В данной работе мы изучаем подполугруппы свободных леворегулярных
Дата поступления полугрупп. Будет показано, что не любая правонаследственная полугруппа вкладыва-
11.03.2018 ется в свободные леворегулярные полугруппы.
Дата принятия в печать 29.03.2018
Дата онлайн-размещения 29.10.2018
Ключевые слова
Леворегулярные полугруппы, правая наследственность
Финансирование
Исследование выполнено при поддержке гранта Российского фонда фундаментальных исследований в рамках научного проекта № 18-31-00330
ON SUBSEMIGROUPS OF FREE LEFT REGULAR BANDS
(paper dedicated to the memory of professor Georgii Petrovich Kukin)
А. N. Shevlyakov
Sobolev Institute of Mathematics SB RAS, Omsk Branch, Omsk, Russia
Article info Abstract. We study subsemigroups of free left regular bands and prove that the right he-
Received reditary property is not sufficient for an embedding into free left regular bands.
11.03.2018
Accepted 29.03.2018
Available online 29.10.2018
Keywords
Left regular bands, right hereditary property
Acknowledgements
The reported study was funded by Russian Fund of Fundamental Research according to the research project № 18-31-00330
1. Introduction
The study of subalgebras of free algebras is a popular branch of contemporary algebra. In the current pa-
per we study this problem in the variety of left regular bands. Left regular bands have important applications in
Вестник Омского университета 2018. Т. 23, № 3. С. 56-58
ISSN 1812-3996-
hyperplane arrangements, random walks and matroid theory, see [1-7].
S. Margolis, F. Saliola, B. Steinberg [4] give the definition of a right hereditary semigroup and show that any subsemigroup of a free left regular band is right hereditary. There naturally arises the question: is any right hereditary left regular band embedded into a free left regular band? Moreover, what are epy sufficient conditions for embedding into free left regular bands?
When I was in Israel in 2016, I discussed these problems with S. Margolis. He recommended me to find a right hereditary semigroup which is not embeddable into free left regular bands. The aim of the current paper is to define this example.
A semigroup S is a left regular band if the next identities
2
X2 = x, xyx = xy
hold in S.
Remark that we always consider left regular bands with the identity e adjoined, i.e. se = es = s for each s E S. Let Fn be the free left regular band of rank n. The elements of Fn are all words w in the alphabet {a1,a2,... ,an} such that any letter at occurs at most one time in w. The product of two elements w1,w2 E Fn is defined as follows:
w1w2 = W1° (w2)3, where ° is the word concatenation and the operator 3 is the deletion of all letters which occur earlier. For example, (a1a2a3)(a2a3a4) = a1a2a3a4.
Below elements of Fn (free generators at) are called words (respectively, letters), and we denote the set of all letters of a word w by L(w).
For elements x,y of a left regular band S one can define a relation
X < y O xy = y.
It is easy to check that < is a partial order over S. Remark that in some papers (e.g. in [4]) < defines the contrary relation: yx=x. For elements x,y E Fn the relation x < y means that a word x is a prefix of y.
Following [4], a left regular band S is right hereditary if the Hasse diagram of the order < is a tree. Obviously, any subsemigroup of a free left regular band is right hereditary, but results below show that there exists a finite right hereditary left regular band S such that S is not embeddable into any free left regular band.
Lemma. An equality x1x2 = y1y2 implies either x1 < y1 or yL < x1 in any right hereditary left regular band S.
Proof. Indeed, x1x2 = y1y2 > y1 and x1x2 > . By the left hereditary property, we obtain that either
x1 < y1 or y1 < x1.
An element z E S is a left zero of a semigroup S if zs=z for any s E S. For example, a word w E Fn is a left zero iff w contains all letters {a1, a2,..., an}. Moreover, if w1w2 = w1 for w1,w2 E Fn then L(w2) £ L(w1).
Below we define a finite left regular band S such
that:
1. S is right hereditary;
2. S is not embeddable into Fn for any natural n. Let S be the union of the free left regular band
F3 = { a1, a2, a3) and a new element z such that:
1. zf = z for any f E S (i. e. z is a left zero of S);
2. z acts on F3 by right multiplications as follows
a1z = a1a2a3,
a2Z = a2a3ai,
a?z = a^a, a
— u.3u.^2-
We have that z acts on other elements of F3 as fol-
lows.
Table 1
f^F3 fz
ala2 ага2а3
o2a3a1
a3ai 030^2
ala3 aiÜ3Ü2
a2al а2а10з
0-30.2 0.30.20.1
for any left zero f £ F3 f
One can directly check that (f1f2)z = f1(f2z) for any f-i, f2 E F3, and therefore S is a semigroup. It is easy to prove that S is a left regular band.
The Hasse diagram of the order < in S is the following.
Fig. 1
Thus, S is a right hereditary left regular band. Theorem. The semigroup S is not embeddable into any free left regular band.
Proof. Assume there is an embedding S ^ Fn for some n. Since a1,a2,a3 are free generators of F3 c s, the sets
Вестник Омского университета 2018. Т. 23, № 3. С. 56-58
-ISSN 1812-3996
L! = L(4>(aj) \ (Lfrfa)) U L(<p(a3))), L2 = L(<p(a2)) \ (Lfriaj) U L(<p(a3))), L3 = L(cp(a3)) \ (L(^(ai)) U L(<p(a2))) are non-empty (if L3 = 0 then <p{ai)<p{a2)<p{a3) = $(ai)$(a2) ^ Q^ia^) = = $(0-10.2) ^ ai Œ2Œ3 = a.ia.2, and we obtain a contradiction).
Since z is a left zero, Lt £ L(<p(z)). Without loss of generality, we put $(z) = w0piwip2w2p3w3, where pt £ Lt and L(w0) n (Li U L2U L3) = 0, LÇwJ n (L2 U L3) = 0 , L(w2) n L3 = 0.
The equality a2z = a2a3a1 becomes $(a.2)woP1w1P2W2P3w3 =
Since Fn is right hereditary, our Lemma provides the words $(a2)w0pi , $(a2)$(a3) are <-comparable. Since the word $(a2)$(a3) does not contain pi, we have $(a2)$(a3) < a2)w0pi. However, the letter P3 E L($(a3)) £ L($(a2)$(a3)) does not occur in the word a2)w0pi, and we obtain a contradiction with the assumption.
REFERENCES
1. Brown K. Semigroups, rings, and Markov chains // J. Theoret. Probab. 2000. Vol 13(3). P. 871-938.
2. Margolis S., Saliola F., Steinberg B. Poset topology and homological invariants of algebras arising in algebraic combinatorics// arXiv:1311.6680.
3. Margolis S., Saliola F., Steinberg B. Semigroups embeddable in hyperplane monoids // Semigroup Forum. 2014. Vol. 89(1). P. 236-248.
4. Margolis S., Saliola F., Steinberg B. Combinatorial topology and the global dimension of algebras arising in combinatorics // J. Eur. Math. Soc. 2015. Vol. 17(12). P. 3037-3080.
5. Saliola F. The quiver of the semigroup algebra of a left regular band // Internat. J. Algebra Comput. 2007. Vol. 17(8). P. 1593-1610.
6. Saliola F. The face semigroup algebra of a hyperplane arrangement // Canad. J. Math. 2009. Vol. 61(4). P. 904-929.
7. Saliola F. The quiver of the semigroup algebra of a left regular band // Int. J. Algebra Comput. 2007. Vol. 17(8). P. 1593-1610.
ИНФОРМАЦИЯ ОБ АВТОРЕ
Шевляков Артём Николаевич - доктор физико-математических наук, старший научный сотрудник, Институт математики им. С. Л. Соболева Сибирского отделения Российской академии наук, Омский филиал, 644043, Россия, г. Омск, ул. Певцова, 13; e-mail: e-mail:a_shevl@mail.ru.
ДЛЯ ЦИТИРОВАНИЯ
Шевляков А. Н. О подполугруппах свободных лево-регулярныхных полугрупп // Вестн. Ом. ун-та. 2018. Т. 23, № 3. С. 56-58. DOI: 10.25513/1812-3996.2018. 23(3).56-58.
INFORMATION ABOUT THE AUTHOR
Shevlyakov Artem Nikolaevich - Doctor of Physical and Mathematical Sciences, Senior Researcher, Sobolev Institute of Mathematics Siberian Branch of the Russian Academy of Sciences, Omsk Branch, 13, Pevtsova st., Omsk, 644099, Russia; e-mail: a_shevl@mail.ru.
FOR QTATIONS
Shevlyakov A.N. On subsemigroups of free left regular bands. Vestnik Omskogo universiteta = Herald of Omsk University, 2018, vol. 23, no. 3, pp. 56-58. DOI: 10.25513/1812-3996.2018.23(3).56-58. (in Russ.).
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