Lattices of classes of groupoids with one-sided quasigroup conditions

Lattices of classes of groupoids with one-sided quasigroup conditions

It is shown that two classes of groupoids satisfying certain one-sided quasigroup conditions, namely the classes of one-sided torsion groupoids and of one-sided finite exponent groupoids, are complete lattices, both isomorphic to the lattice of Steinitz numbers with the divisibility relation.

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1. Verfasser: Galuszka, J.
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spelling irk-123456789-1548032019-06-17T01:30:33Z Lattices of classes of groupoids with one-sided quasigroup conditions Galuszka, J. It is shown that two classes of groupoids satisfying certain one-sided quasigroup conditions, namely the classes of one-sided torsion groupoids and of one-sided finite exponent groupoids, are complete lattices, both isomorphic to the lattice of Steinitz numbers with the divisibility relation. 2010 Article Lattices of classes of groupoids with one-sided quasigroup conditions / J. Galuszka// Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 31–40. — Бібліогр.: 13 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:05B15, 08A30, 08A40, 08A62,08A99, 14R10, 20N02, 20N05. http://dspace.nbuv.gov.ua/handle/123456789/154803 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 It is shown that two classes of groupoids satisfying certain one-sided quasigroup conditions, namely the classes of one-sided torsion groupoids and of one-sided finite exponent groupoids, are complete lattices, both isomorphic to the lattice of Steinitz numbers with the divisibility relation.
format Article
author Galuszka, J.
spellingShingle Galuszka, J.
Lattices of classes of groupoids with one-sided quasigroup conditions
Algebra and Discrete Mathematics
author_facet Galuszka, J.
author_sort Galuszka, J.
title Lattices of classes of groupoids with one-sided quasigroup conditions
title_short Lattices of classes of groupoids with one-sided quasigroup conditions
title_full Lattices of classes of groupoids with one-sided quasigroup conditions
title_fullStr Lattices of classes of groupoids with one-sided quasigroup conditions
title_full_unstemmed Lattices of classes of groupoids with one-sided quasigroup conditions
title_sort lattices of classes of groupoids with one-sided quasigroup conditions
publisher Інститут прикладної математики і механіки НАН України
publishDate 2010
url http://dspace.nbuv.gov.ua/handle/123456789/154803
citation_txt Lattices of classes of groupoids with one-sided quasigroup conditions / J. Galuszka// Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 31–40. — Бібліогр.: 13 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT galuszkaj latticesofclassesofgroupoidswithonesidedquasigroupconditions
first_indexed 2025-07-14T06:53:32Z
last_indexed 2025-07-14T06:53:32Z
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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 Volume 9 (2010). Number 1. pp. 31 – 40 c© Journal “Algebra and Discrete Mathematics” Lattices of classes of groupoids with one-sided quasigroup conditions Jan Ga luszka Communicated by V. I. Sushchansky Abstract. It is shown that two classes of groupoids satisfying certain one-sided quasigroup conditions, namely the classes of one- sided torsion groupoids and of one-sided finite exponent groupoids, are complete lattices, both isomorphic to the lattice of Steinitz numbers with the divisibility relation. 1. Introduction Recall that Steiner quasigroups can be defined as groupoids which are idempotent commutative and satisfy xy2 = x where xy2 denotes the term (xy)y. So-called ‘power conditions’ (like (∗n) and (m∗) in Section 1.2) are quite often applied in group and groupoid theory (see for example [4]). Groupoids satisfying at least one of (∗n) and (m∗) are one-sided quasi- groups (the terminology is briefly recalled in Section 1.1). In [6] it is shown among other things that the varieties of groupoids defined by (∗n) (as well as by (m∗)) form a lattice isomorphic to the lattice of positive integers with the divisibility relation (see [6, Theorem 13], recalled in this paper as Theorem 4). By a slight generalization of conditions (∗n) and (m∗) (see (∗s) and (∗ts) below) we obtain a collection of new classes of groupoids which are also one-sided quasigroups. To describe the algebraic structure formed by these classes we compare it to the lattice structure which arises from an idea of E. Steinitz (see [13, p. 250]). This idea — 2000 Mathematics Subject Classification: 05B15, 08A30, 08A40, 08A62, 08A99, 14R10, 20N02, 20N05. Key words and phrases: Groupoid, quasigroup, right quasigroup, left quasi- group, Steinitz numbers. Jo u rn al A lg eb ra D is cr et e M at h .32 Lattices of classes of groupoids recalled briefly in Section 1.3 — allows us to treat the lattice of positive integers with the divisibility relation as a sublattice of a certain com- plete lattice. Today this idea is known as Steinitz numbers ([2, 12]) (or supernatural or surnatural numbers ([11])). These numbers have found applications in various parts of algebra, especially in field theory, group theory and some related areas. Examples of such applications can be found in [2, 7, 9, 10, 11, 12]. In the present paper it is proved that the classes of one-sided torsion groupoids and of one-sided finite exponent groupoids (the definitions are recalled in Section 1.2) form lattices both isomorphic to the lattice of Steinitz numbers with the divisibility relation. This main result is presented in Section 2 (Theorem 7). Parts of the results of this paper were announced without proof at the AAA 76 – 76th International Workshop on General Algebra (Linz 2008). 1.1. Main notation and notions The paper is closely connected with [5] and [6], so similar terminology and notation is used. For convenience of the reader we repeat some material from [5] and [6] without proofs, making our presentation self-contained. All undefined notions and notations are standard and can be found in [3] or [8]. Some basic notions specific to quasigroup theory are analogous to those introduced in [1]. We also use the following notations and terminology: • N denotes the set of nonnegative integers, Nk def = N− {0, . . . , k − 1}. • If n,m ∈ N1, then n | m means that n divides m. • If A ⊆ CnCnCn is a set of cardinal numbers, then lcm(A) and gcd(A) denote the least common multiple and the greatest common divisor of the set A respectively. • If A and B are sets, then AB def = {f | f : B −→ A} denotes the set of all maps from B to A. By a groupoid is meant a pair G = (G, · ) with universe (base set) G and binary operation · : G×G −→ G ((x, y) 7→ xy). In the following, the symbol G stands for groupoids only. Recall that in groupoid theory, by a right (resp. left) quasigroup we mean a groupoid G such that for all a, b ∈ G the equation xa = b (resp. ax = b) has a unique solution. By a quasigroup we mean a groupoid which is a right and left quasigroup simultaneously. For a groupoid G = (G, · ) we have the dual groupoid G← = (G, ◦ ) where x ◦ y def = yx. Clearly Jo u rn al A lg eb ra D is cr et e M at h .J. Ga luszka 33 (G←)← = G. Let t be a term over a language appropriate for groupoid theory. Let G be a groupoid. Then the interpretation tG ← is named the dual sentence to the interpretation tG. Thus, if a groupoid G is a right quasigroup then its dual groupoid is a left quasigroup and vice versa. This duality establishes a symmetrical correspondence between ‘right’ and ‘left’ versions of statements (to every theorem in the right version corresponds its dual left version and vice versa). Therefore, for concise- ness we formulate almost all statements below in one (right) version only. The term ‘one-sided finite exponent groupoid’ will mean a finite right exponent groupoid or a finite left exponent groupoid (the definitions are recalled in Section 1.2). The term ‘one-sided torsion groupoid’ is under- stood analogously. 1.2. Classes of groupoids with right (left) quasigroup proper- ties Let us recall some idea presented in [6]. The family of ‘power’ terms {xyn | n ∈ N1} is inductively defined as follows: xy1 = xy, xyn = (xyn−1)y, and similarly for {nyx | n ∈ N1}. With these families of terms there are naturally associated some families of identities. For n,m ∈ N1 we have the following ‘power’ identities: xyn = x, (∗n) myx = x. (m∗) Let Qn (resp. mQ) denote the variety of groupoids defined by the identity (∗n) (resp. (m∗)). Moreover mQn def = mQ ∩Qn. The elements of Qn will be called groupoids of right exponent n. Let Q def = {Qn | n ∈ N1}. Set Q∗ def = ⋃ Q. A groupoid G is said to be of finite right exponent if G ∈ Q∗. The varieties Qn (n ∈ N1) and the class Q∗ were studied in [6]. Lemma 1. ([6, Lemma 1])Let G be a groupoid, and a, b ∈ G. (i) If abn = a and k ∈ N1, then abkn = a. (ii) If abn = a and abm = a, then abgcd(m,n) = a. (iii) If abn = a and kba = a, then ablcm(k,n) = a and lcm(k,n)ba = a. Proposition 2. ([6, Proposition 2]) For n, k ∈ N1 the following state- ments hold: (i) Qn ⊆ Qkn; Jo u rn al A lg eb ra D is cr et e M at h .34 Lattices of classes of groupoids (ii) Qk ∪ Qn ⊆ Qm, where m = lcm(k, n); (iii) Qk ∩ Qn = Qd, where d = gcd(k, n); (iv) kQn ⊆ mQm, where m = lcm(k, n). Proposition 3. ([6, Proposition 9]) (i) If G satisfies (∗n) for some n ∈ N1, then G is a right quasigroup. (ii) If G satisfies both (∗n) and (m∗) for some n,m ∈ N1, then G is a quasigroup. Theorem 4. ([6, Theorem 13]) The lattice Q∗ = (Q, ∧,∨) is isomorphic to N1 = (N1, gcd, lcm). The following formulas can be seen as a natural generalization of the identities (∗n) (cf. [6]): ∀x, y ∃n ∈ N1 xyn = x. (∗t) We call G right-torsion if the formula (∗t) is satisfied in G. The class of right-torsion groupoids is denoted by T ∗. Denote by QG∗ (resp. ∗QG) the class of right (resp. left) quasigroups. Proposition 5. ([6, Proposition 14]) Q∗ ( T ∗ ( QG∗. Clearly neither Q∗ nor T ∗ is a variety. 1.3. Steinitz numbers We recall briefly an idea of extension of positive integers introduced by E. Steinitz. The construction and notation we propose are presented in the form suitable for our further considerations. Let P be the set of prime numbers. There exists a 1-1 map α : N1 −→ S◦ (n 7→ α(n)), (1) where S◦ = {x | x ∈ NP and |{p | xp 6= 0}| < ℵ0}, i.e. S◦ is the set of maps from P to N which are zero almost everywhere. The mapping α(n) is defined as the infinite sequence α(n) = 2α(n)23α(n)35α(n)5 . . . pα(n)p . . . Jo u rn al A lg eb ra D is cr et e M at h .J. Ga luszka 35 which corresponds to the unique prime factorization of n. Thus we can write slightly informal equalities n = ∏ p∈P, α(n)p 6=0 pα(n)p = α(n) = 2α(n)23α(n)35α(n)5 . . . pα(n)p . . . (2) identifying these formally different objects. Let us consider the following two lattices: • N1 = (N1, |), the lattice of positive integers with the divisibility rela- tion, • S◦ = (S◦,≤) where ≤ denotes the product relation, i.e. for a, b ∈ S◦ we have a ≤ b if and only if ap ≤ bp for every p ∈ P. It is clear that n | m ⇐⇒ α(n) ≤ α(m), (3) so that α : N1 iso −→ S ◦, (4) i.e. α is an isomorphism of lattices. Using the traditional intuitive and slightly informal notation and ap- plying the identification (2) we write n = α(n) = ∏ p∈P pα(n)p (5) where in the formal symbol ∏ p∈P p α(n)p almost all factors are equal to 1. Using this notation we can write gcd(n,m) = min{α(n), α(m)} = ∏ p∈N pmin{α(n)p,α(m)p}, (6) lcm(n,m) = max{α(n), α(m)} = ∏ p∈N pmax{α(n)p,α(m)p}. (7) Let N̄ = N ∪ {ω} be an extension of N by adding a new element ω. We extend the standard relation ≤ by defining n ≤ ω for all n ∈ N̄. Thus N̄ = (N̄,≤) is a bounded chain. Define S = N̄P. Then S is called the set of Steinitz numbers. Thus S = (S,≤) with the product relation ≤ is a complete (and evidently bounded) lattice called the lattice of Steinitz numbers. It is clear that S◦ is a sublattice of S. For Steinitz numbers we introduce the notation analogous to (5). If s ∈ S then we write s = ∏ p∈P psp (8) Jo u rn al A lg eb ra D is cr et e M at h .36 Lattices of classes of groupoids where sp ∈ N̄ and it is not assumed that almost all factors are equal to 1. On S one can define the divisibility relation | as follows. For r, s ∈ S, s | r def ⇐⇒ s ≤ r. Thus the map α is an embedding of the lattice N1 into S = (S, |) (see (4)). Statements (9) and (10) below can be seen as the analogues of (6) and (7) respectively for Steinitz numbers. Let A ⊆ S and A 6= ∅. Then gcd(A) = min{s | s ∈ A} = ∏ p∈N pmin{sp|s∈A}, (9) lcm(A) = max{s | s ∈ A} = ∏ p∈N pmax{sp|s∈A}. (10) S as a relational system determines the algebra Sa = (S, gcd, lcm, 1, ω) of type (2, 2, 0, 0) which is a bounded lattice in algebraic interpretation. Some examples The positive integer 30 can be interpreted as the Steinitz number 213151 (factors equal to 1, i.e. of the form p0, are omitted). Numbers like 2ω3151 or ∏ p∈N p have no interpretation as positive integers. For every n ∈ N we have 2n3151 | 2ω3151. Moreover, ω = ∏ p∈N pω and 1 = ∏ p∈N p0 are the maximum and minimum in S respectively. 2. Classes of groupoids with one-sided quasigroup condi- tions The notion of Steinitz numbers allows us to present the classes of finite one-sided exponent groupoids and of one-sided torsion groupoids as col- lections of subclasses satisfying so-called bounded conditions. Let s ∈ S. The following formulas can be seen as a generalization of the identities (∗n) for Steinitz numbers: ∃n ∈ N1 ∀x, y n | s, xyn = x. (∗s) Thus G satisfies (∗s) if and only if there exists n ∈ N1 such that n | s and G satisfies (∗n). Let Qs̄ be the class of groupoids satisfying (∗s). The members of Qs̄ will be called s-bounded right exponent groupoids. Clearly (for example by Proposition 2(i)), if m ∈ N1 then Qm = Qm̄. This identity allows us to simplify notation by omitting the bar over s in Qs̄. In this terminology (clearly Q∗ = Qω), the class of finite right Jo u rn al A lg eb ra D is cr et e M at h .J. Ga luszka 37 exponent groupoids is the class of ω-bounded right exponent groupoids, and the variety Q1 (i.e. the variety 000∗ of left-zero groupoids) is the class of 1-bounded right exponent groupoids. Clearly if s ∈ S then Qs = ⋃ n∈N1, n|s Qn. (11) As a generalization of the condition (∗t) we have: ∀x, y ∃n ∈ N1 n | s, xyn = x. (∗ts) Let Ts be the class of all groupoids satisfying (∗ts). The elements of Ts will be called s-bounded right torsion groupoids. Proposition 6. (i) If m ∈ N1, then Qm = Tm. (ii) Let s1, s2 ∈ S be such that s1 | s2. Then Qs1 ⊆ Qs2 and Ts1 ⊆ Ts2 . (iii) If s ∈ S, then Qs ⊆ Ts. If s ∈ S and s is not a positive integer, then Qs ( Ts. Proof. (i) By the definitions of Qm and Tm, clearly Qm ⊆ Tm. Let G ∈ Tm. Then for every a, b ∈ G there exists n ∈ N such that n | m and abn = a. By Lemma 1 we see that for every a, b ∈ G the equality abm = a is satisfied. Hence the identity xym = x is satisfied in G and G ∈ Qm. (ii) This is an easy consequence of the definition of Qs and Ts for s ∈ S. (iii) It is clear that Qs ⊆ Ts. Assume that s is not a positive integer. To prove that Qs 6= Ts we use a construction similar to the one in [6, Example 8]. By assumption, s is a Steinitz number that is not a positive integer. Thus s = ∏ p∈P p sp and there are two mutually not necessarily exclusive possibilities: (a) There exists p ∈ P such that sp = ω, (b) The set {sp | sp 6= 0} is not finite. In case (a), fix p ∈ P such that sp = ω. Let f = (fn)n∈N1 be a family of permutations of N1 such that fn def = (1 . . . np) ∪ idNnp+1 i.e. fn is a cyclic permutation of {1, . . . , np} and the identity elsewhere. Let Nf = (N1, ·f ) with a ·f b = fb(a). In the following we omit the symbol ·f in products. Evidently the groupoid Nf satisfies condition (∗tpω). Thus Nf ∈ Tpω . It is clear that pω | s. Therefore Tpω ⊆ Ts by Jo u rn al A lg eb ra D is cr et e M at h .38 Lattices of classes of groupoids (ii). Hence Nf ∈ Ts. Suppose that Nf satisfies condition (∗s). Then Nf satisfies the identity xym = x for some m ∈ N1 such that m | s. Evidently α(m)p < ω. Let b > α(m)p. Then bp > α(m)p. Therefore 1bbp = 1 (by the construction of Nf ) and 1bm = 1 (by the assumption that Nf satisfies the identity xym = x). From Lemma 1 (ii) we have 1 = 1bgcd(bp,m) = 1bα(m)p , a contradiction. Hence Nf ∈ Ts −Qs̄. In case (b), there exists an infinite sequence p̄ = (p1, p2 . . .) of prime numbers such that spn ≥ 1 for every n ∈ N1. Let f = (fn)n∈N1 be a family of permutations of N1 such that fn def = { (1 . . . pi) ∪ idNpi+1 if n = pi for some i ∈ N1, idN1 otherwise. As in the previous item, let Nf = (N1, ·f ) with a ·f b = fb(a). Thus Nf ∈ Ts′ where s′ = ∏ p∈P p α(s′)p is such that α(s′)p = { 1 if p = pi for some i ∈ N1, 0 otherwise. We have Nf ∈ Ts, because Nf ∈ Ts′ and s′ | s (see (ii)). Suppose that Nf satisfies condition (∗s). Then Nf satisfies the identity xym = x for some m ∈ N1 such that m | s. Let pi be a prime number appearing in the sequence p̄ and not appearing in the prime decomposition of m. We have 1ppii = 1 and 1pmi = 1. Therefore 1 = 1p gcd(pi,m) i = 1pi by Lemma 1 (ii), a contradiction. Thus Nf ∈ Ts −Qs̄. Let Q̄ def = {Qs | s ∈ S} and T̄ def = {Ts | s ∈ S}. We have Q∗ = Qω = ⋃ Q̄ and T ∗ = Tω = ⋃ T̄ . Theorem 7. Q̄∗ = (Q̄,⊆) and T̄∗ = (T̄ ,⊆) are complete lattices, both isomorphic to the lattice of Steinitz numbers S = (S, |). Proof. Let us consider the relational system Q̄∗ = (Q̄,⊆) and the com- plete lattice S = (S, |). Let us take the map ϕQ : S −→ Q̄ (x 7→ ϕQ(x) def = Qx). We will prove that ϕQ : S iso −→ Q̄∗ (i.e. ϕQ is an isomorphism of relational systems). The proof of this fact is divided into two steps: (a) ϕQ is a bijection, (b) ϕQ and ϕ−1Q are homomorphisms of relational systems. Jo u rn al A lg eb ra D is cr et e M at h .J. Ga luszka 39 To prove (a), let s1, s2 ∈ S, s1 6= s2. Without loss of generality we can assume that there exists a prime number p such that s1,p < s2,p. Thus there exists a positive integer s′2,p such that s1,p < s′2,p ≤ s2,p. Let Nf = (N1, ·f ) with fn def = { (1 . . . s′2,p) ∪ idN s′ 2,p +1 if n = p, idN1 otherwise. Thus Nf ∈ Qs2 −Qs1 . Therefore ϕQ(s1) = Qs1 6= Qs2 = ϕQ(s2). It is obvious that ϕQ is a surjection. To prove (b), let s1, s2 ∈ S with s1 | s2. By Proposition 6 (ii), ϕQ(s1) = Qs1 ⊆ Qs2 = ϕQ(s2). Now assume that Qs1 ⊆ Qs2 and suppose that s1 ∤ s2. Then s2,p < s1,p for some prime number p. Thus there exists a positive integer s′1,p such that s2,p < s′1,p ≤ s1,p. Let Nf = (N1, ·f ) with fn def = { (1 . . . s′1,p) ∪ idN s′ 1,p +1 if n = p, idN1 otherwise. Thus Nf ∈ Qs1 −Qs2 , a contradiction. Using similar arguments one can prove that ϕT : S −→ T̄ (x 7→ ϕT (x) def = Tx) is an isomorphism of the relational systems S and T̄∗. The complete lattices Q̄∗, T̄∗ as relational systems determine the alge- bras Q̄∗a = (Q̄,∧,∨,000∗,Q∗) and T̄∗a = (T̄ ,∧,∨,000∗, T ∗) respectively, both of type (2, 2, 0, 0); these are bounded lattices in algebraic interpretation. As an immediate consequence of Theorem 7 we obtain the following corol- lary: Corollary 8. Q̄∗a ≃ T̄∗a ≃ Sa. References [1] V.D. Belousov, Foundations of the Theory of Quasigroups and Loops (Russian), Izdat. “Nauka”, Moscow, 1967. [2] J.V. Brawley and G.E. Schnibben, Infinite Algebraic Extensions of Finite Fields, Contemp. Math. 19, Amer. Math. Soc., Providence, RI, 1989. [3] S. Burris and H. P. Sankappanavar, A course in universal algebra, Springer, New York, 1981. Jo u rn al A lg eb ra D is cr et e M at h .40 Lattices of classes of groupoids [4] J. Dudek and A. Kisielewicz, Totally commutative semigroups, J. Austral. Math. Soc. Ser. A 51 (1991), no. 3, 381–399. [5] J. Ga luszka, Codes of groupoids with one-sided quasigroup conditions, Algebra Discrete Math. (2009), no. 2, 27–44. [6] J. Ga luszka, Groupoids with quasigroup and Latin square properties, Discrete Math. 308 (2008), no. 24, 6414–6425. [7] R. Gilmer, Zero-dimensional subrings of commutative rings, in Abelian groups and modules (Padova, 1994), 209–219, Kluwer Acad. Publ., Dordrecht. [8] G. Grätzer, Universal Algebra, Van Nostrand, 1979. [9] N. V. Kroshko and V. I. Sushchansky, Direct limits of symmetric and alternating groups with strictly diagonal embeddings, Arch. Math. (Basel) 71 (1998), no. 3, 173–182. [10] J. S. Richardson, Primitive idempotents and the socle in group rings of periodic abelian groups, Compositio Math. 32 (1976), no. 2, 203–223. [11] A. Robinson, Nonstandard arithmetic, Bull. Amer. Math. Soc. 73 (1967), 818– 843. [12] S. Roman, Field Theory, Springer-Verlag, 1995 [13] E. Steinitz, Algebraische Theorie der Körper, J. reine angew. Math. 137 (1910), 167–309. Contact information J. Ga luszka Institute of Mathematics, Silesian Univer- sity of Technology, Kaszubska 23, 44-100 Gliwice, Poland E-Mail: jan.galuszka@polsl.pl Received by the editors: 26.11.2009 and in final form 26.11.2009.

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