Idempotent D -cross-sections of the finite inverse symmetric semigroup ISn

Idempotent D -cross-sections of the finite inverse symmetric semigroup ISn

We prove that every finite poset can be embedded in some idempotent D-cross-section of the finite inverse symmetric semigroup ISn.

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Datum:2008
1. Verfasser: Pyekhtyeryev, V.
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Sprache:English
Veröffentlicht: Інститут прикладної математики і механіки НАН України 2008
Schriftenreihe:Algebra and Discrete Mathematics
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Zitieren:Idempotent D -cross-sections of the finite inverse symmetric semigroup ISn / V. Pyekhtyeryev // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 3. — С. 84–97. — Бібліогр.: 7 назв. — англ.

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spelling irk-123456789-1533592019-06-15T01:26:27Z Idempotent D -cross-sections of the finite inverse symmetric semigroup ISn Pyekhtyeryev, V. We prove that every finite poset can be embedded in some idempotent D-cross-section of the finite inverse symmetric semigroup ISn. 2008 Article Idempotent D -cross-sections of the finite inverse symmetric semigroup ISn / V. Pyekhtyeryev // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 3. — С. 84–97. — Бібліогр.: 7 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20M20, 20M10. http://dspace.nbuv.gov.ua/handle/123456789/153359 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 We prove that every finite poset can be embedded in some idempotent D-cross-section of the finite inverse symmetric semigroup ISn.
format Article
author Pyekhtyeryev, V.
spellingShingle Pyekhtyeryev, V.
Idempotent D -cross-sections of the finite inverse symmetric semigroup ISn
Algebra and Discrete Mathematics
author_facet Pyekhtyeryev, V.
author_sort Pyekhtyeryev, V.
title Idempotent D -cross-sections of the finite inverse symmetric semigroup ISn
title_short Idempotent D -cross-sections of the finite inverse symmetric semigroup ISn
title_full Idempotent D -cross-sections of the finite inverse symmetric semigroup ISn
title_fullStr Idempotent D -cross-sections of the finite inverse symmetric semigroup ISn
title_full_unstemmed Idempotent D -cross-sections of the finite inverse symmetric semigroup ISn
title_sort idempotent d -cross-sections of the finite inverse symmetric semigroup isn
publisher Інститут прикладної математики і механіки НАН України
publishDate 2008
url http://dspace.nbuv.gov.ua/handle/123456789/153359
citation_txt Idempotent D -cross-sections of the finite inverse symmetric semigroup ISn / V. Pyekhtyeryev // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 3. — С. 84–97. — Бібліогр.: 7 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT pyekhtyeryevv idempotentdcrosssectionsofthefiniteinversesymmetricsemigroupisn
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fulltext Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 3. (2008). pp. 84 – 87 c© Journal “Algebra and Discrete Mathematics” Idempotent D-cross-sections of the finite inverse symmetric semigroup ISn Vasyl Pyekhtyeryev Communicated by V. V. Kirichenko Abstract. We prove that every finite poset can be embedded in some idempotent D-cross-section of the finite inverse symmetric semigroup ISn. The symmetric group Sn is a central object of study in many branches of mathematics. There exist several "natural" analogues (or generaliza- tions) of Sn in semigroup theory. The most classical ones are the sym- metric semigroup Tn and the inverse symmetric semigroup ISn. They arise when one tries to generalize Cayley’s Theorem to the classes of all semigroups or all inverse semigroups respectively. A less obvious semi- group generalizations of Sn is the so-called Brauer semigroup Bn, which appears in the context of centralizer algebras in representation theory, see [1]. Let n be a positive integer. Let us put N = {1, . . . , n} and N ′ = {1′, . . . , n′}. The elements of the Brauer semigroup Bn are all possible partitions of the set N ∪ N ′ into two-element blocks. Consider the map ′ : N → N ′ as a fixed bijection and denote the inverse bijection by the same symbol, i. e. (a′)′ = a for all a ∈ N . For α ∈ Bn and two different elements a, b ∈ N ∪ N ′ we set a ≡α b provided that {a, b} ∈ α. In other words, ≡α is the equivalence relation corresponding to the partition α. Let α = X1 ∪ . . . ∪ Xn and β = Y1 ∪ . . . ∪ Yn be two elements from Bn. Let us define a new equivalence relation, ≡, on N ∪ N ′ as follows: 2000 Mathematics Subject Classification: 20M20, 20M10. Key words and phrases: Inverse symmetric semigroup, Green’s relations, cross-sections, posets. Jo u rn al A lg eb ra D is cr et e M at h .V. Pyekhtyeryev 85 • for a, b ∈ N we have a ≡ b if and only if a ≡α b or there is a sequence, c1, . . . , c2s, s ≥ 1, of elements of N such that a ≡α c′1, c1 ≡β c2, c′2 ≡α c′3, . . . , c2s−1 ≡β c2s and c′2s ≡α b. • for a, b ∈ N we have a′ ≡ b′ if and only if a′ ≡β b′ or there is a sequence, c1, . . . , c2s, s ≥ 1, of elements of N such that a′ ≡β c1, c′1 ≡α c′2, c2 ≡β c3, . . . , c′2s−1 ≡α c′2s and c2s ≡β b′. • for a, b ∈ N we have a ≡ b′ if and only if b′ ≡ a if and only if there is a sequence, c1, . . . , c2s−1, s ≥ 1, of elements of N such that a ≡α c′1, c1 ≡β c2, c′2 ≡α c′3, . . . , c′2s−2 ≡α c′2s−1 and c2s ≡β b′. It is easy to see that ≡ determines a partition of N ∪N ′ into two-element subsets and so belongs to Bn. We define this element to be the product αβ. Thus, the study of the structure of these semigroups is a natural problem to investigation. Let ρ be an equivalence relation on a semigroup S. A subsemigroup T ⊂ S is called a cross-section with respect to ρ if T contains exactly one element from every equivalence class. Clearly, the most interesting are the cross-sections with respect to the equivalence relations connected with the semigroup structure on S. The first candidates for such relations are congruences and the Green’s relations, which are important tools in the description and decomposition of semigroups. For any a ∈ S we denote by L(a) (R(a), J(a)) the principal left (right, two-sided) ideal generated by a respectively. The Green’s relations L, R, H, D and J on semigroup S are defined as binary relations in the following way: aLb if and only if L(a) = L(b); aRb if and only if R(a) = R(b); aJ b if and only if J(a) = J(b) for any a, b ∈ S and the relation H = L ∩ R, while the relation D = L ∨ R, where the join is in the lattice of all equivalences on S, that is D is the least equivalence containing both L and R. Cross-sections with respect to congruences are called retracts. They are important in study of semigroup endomorphisms. Cross-sections with respect to the H- (L-, R-, D-, J -) Green’s rela- tions are called H- (L-, R-, D-, J -) cross-sections in the sequel. During the last decade cross-sections of Green’s relations for some classical semigroups were studied by different authors. In particular, for the inverse symmetric semigroup ISn all H-cross-sections were classified in [2] and all L- and R-cross-sections were classified in [3]. For the infinite inverse symmetric semigroup ISX all H-, L- and R-cross-sections were classified in [7], and for the symmetric semigroup TX all H- and R-cross- Jo u rn al A lg eb ra D is cr et e M at h .86 Idempotent D−cross-sections of ISn sections were classified in [5], [6]. The classification of H-, L- and R- cross-sections for the Brauer semigroup Bn was obtained in [4]. The problem of classification of D-cross-sections for these semigroups is essentially more difficult, since every D-class has large cardinality and so the semigroups have many different D-cross-sections. We consider idempotent D-cross-sections of the finite symmetric in- verse semigroup ISn on the set N = {1, . . . , n}, that is cross-sections which consist of idempotents. For the first time the problem of classifi- cation of these cross-sections appeared in [3] and it is still open. Let us recall that every idempotent of ISn has the form idA, where A ⊆ N and Green’s D-classes are Dk = { a ∈ ISn | rk(a) = k}, 0 ≤ k ≤ n. Hence one can naturally construct a partial order on the set of all idempotents of this semigroup: idA ≤ idB if and only if A ⊆ B. Thus, one can consider every idempotent D-cross-section of ISn as a poset. Theorem. The boolean of a set M containing exactly n elements is iso- morphic to a some idempotent D-cross-section of the finite symmetric inverse semigroup IS2n−1. Proof. Put M = {0, . . . , n − 1}. Let N be a disjoint union of sets Ni, i = 0, . . . , n − 1, where |Ni| = 2i for every i. Then |N | = 2n − 1. Let us define the map f : 2M → 2N by the rule 2M ∋ K 7→ ⋃ i∈K Ni ∈ 2N . Clearly, the cardinality of the set f(K) equals the integer which binary representation is the boolean vector of the subset K. Therefore all sets from the image of the map f have pairwise different cardinality. More- over, for every number l, 0 ≤ l ≤ 2n −1 there is exists a set K ∈ 2M such that |f(K)| = l. Thus, the subset T = {idf(K) | K ∈ 2M} of the semi- group ISN contains exactly one element from every D-class of this semi- group. Since from equalities idA·idB = idA∩B and f(A)∩f(B) = f(A∩B) we have that the set T is closed under multiplication. Finally, T is an idempotent D-cross-section of ISN , which is isomorphic (as poset) to the boolean of the set M . Remark. The number 2n − 1 in the theorem can not be decreased, be- cause every idempotent D-cross-section of the ISn contains exactly n+1 elements. Corollary. Every finite poset can be embedded in some idempotent D-cross-section of the finite symmetric inverse semigroup ISn. Jo u rn al A lg eb ra D is cr et e M at h .V. Pyekhtyeryev 87 References [1] Brauer R. On algebras which are connected with the semisimple continious groups. Ann. of Math (2) 38, no.4, 1937, pp. 857-872. [2] Cowan D.F., Reilly N. R. Partial cross-sections of symmetric inverse semigroups. Internat J. Algebra Comput. 5, no.3, 1995, pp. 259-287. [3] Ganyushkin O., Mazorchuk V. L− and R−cross-sections in ISn. Com. in Algebra 31, no.9, 2003, pp. 4507-4523. [4] G. Kudryavtseva, V. Maltcev and V. Mazorchuk (2004) L− and R−cross-sections in the Brauer semigroup, Semigroup Forum 72, no.2, 2006, pp. 223-248. [5] Pyekhtyeryev V. H− and R−cross-sections of the full finite semigroup Tn. Algebra and Discrete Mathematics no.3, 2003, pp. 82-88. [6] Pyekhtyeryev V., R−cross-sections of TX Matematychni Studii, no.21, 2004, pp. 133-139. [Ukrainian] [7] Pyekhtyeryev V. H−, R− and L−cross-sections of the infinite symmetric inverse semigroup ISX . Algebra and Discrete Mathematics, no.1, 2005, pp. 92-104. Contact information V. Pyekhtyeryev Department of Mechanics and Mathematics, Kiyv Taras Shevchenko University, 64, Volodymyrska st., 01033, Kiyv, UKRAINE E-Mail: vasiliy@univ.kiev.ua Received by the editors: 05.02.2008 and in final form 14.10.2008.

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