Aczel (1965) investigated quasigroups, 3-nets and nomograms
and Belousov (1971) k-nets and associated (k-1) - quasigroups. There are different 3 - seminets and k-seminets (see e.g. Havel (1967), Taylor (1971), Ušan (1977), Galić (1989), etc.) to which by some rules one can assign corresponding algebraic structures
(partial quasigroups and partial groupoids). Galić (1990) defines
U-k - seminets of the maximal degree and shows the existence and
construction in dependence on the set P over which one constructs
a k-seminet. In this paper it is shown how many U-k - seminets
of maximal degree μ can be constructed over the set P
for the given t-order

R. Galić

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Mathematical Communications 2(1997), 21–25 21The maximal number of U-k – seminetsof the maximal degree∗Radoslav Galić†Abstract. Aczel (1965) investigated quasigroups, 3-nets and nomo-grams and Belousov (1971) k-nets and associated (k-1) – quasigroups.There are different 3 – seminets and k-seminets (see e.g. Havel (1967),Taylor (1971), Ušan (1977), Galić (1989), etc.) to which by some rulesone can assign corresponding algebraic structures (partial quasigroupsand partial groupoids). Galić (1990) defines U-k – seminets of the max-imal degree and shows the existence and construction in dependence onthe set P over which one constructs a k-seminet. In this paper it isshown how many U-k – seminets of maximal degree µ can be constructedover the set P for the given t-order.Key words: U-k – seminets, k – seminets, t – order, maximaldegreeSažetak. Maksimalni broj U-k – polumreža maksimalnogstupnja. Aczel (1965) proučava kvazigrupe, 3 – mreže i nomograme, aBelousov (1971) k – mreže i pridružene (k-1) – kvazigrupe. Postoje ra-zličite 3 – polumreže i k – polumreže (vidi primjerice Havel (1967), Tay-lor (1971), Ušan (1977), Galić (1989) i drugi) kojima se po nekom prav-ilu mogu pridružiti odgovarajuće algebarske strukture (parcijalne kvazi-grupe i parcijalni grupoidi). Galić (1990) definira U-k – polumrežemaksimalnog stupnja te pokazuje postojanje i konstrukciju u ovisnosti oskupu P nad kojim se konstruira k – polumreža. U ovom radu pokazanoje koliko se U-k – polumreža maksimalnog stupnja µ može konstruiratinad skupom P za zadani t-red.Ključne riječi: U-k – polumreže, k – polumreže, t – red, maksi-malni stupanjAMS subject classifications: 51E30, 20N5Received December 23, 1996, Accepted March 4, 1997∗The lecture presented at the Mathematical Colloquium in Osijek organized by the CroatianMathematical Society – Division Osijek, April 26, 1996.†Faculty of Electrical Engineering, Department of Mathematics, Istarska 3, HR-31 000 Osijek,Croatia, e-mail: galic@drava.etfos.hr22 R. GalićLet P be a nonempty set and B family of nonempty subsets of P, i.e. B ⊂P(P)\{∅}. Let Π = {X1, ..., Xk}, k ≥ 3, be a partition of the set B. Elements ofthe set P, B and Π are called points, blocks and classes, respectively. Then theordered triple N = (P,B,Π) is called a U-k – seminet if the following conditionsare satisfied:C1. Any two blocks from different classes have at most one point in common, thatis(∀xi ∈ Xi)(∀xj ∈ Xj)(i 6= j ⇒ |xi ∩ xj | ≤ 1),C2. Every point of P lies in exactly one block from each class, that is(∀p ∈ P)(∀i ∈ {1, 2, ..., k})(∃!xi ∈ Xi : p ∈ xi).If in a U-k – seminet N = (P,B, Π) we need to specify classes, then we willdenote this U-k – seminet by N = (P, X1, ..., Xk), where the number k denotes thedegree of the seminet.Number of points of a set A, i.e. the cardinal number of the set A will bedenoted by |A|.Number of points of a maximal block, i.e. the number:d = max{|b| : b ∈ B}is called t-order of a U-k – seminet.N = (P,B,Π) is said to be a U-µ – seminet of maximal degree µ over theset P if for every U-k – seminet N ′ = (P,B′, Π′) over the set P, one has µ ≥ k.The existence and the construction of a U-k – seminet with maximal degree forsome sets P are proved by Galić (1990). The following theorem considers that issue:Theorem 1. Let P be a set of points such that |P| = t ≥ 3. Then for2 ≤ d < t + 22(1)there exists a U -µ – seminet N = (P,B,Π) of maximal degree µ and t-order equalto d over the set P, so thatµ = t− d + 2. (2)We will consider here maximal degree µ of a U-k – seminet which depends onthe number of points of the set P (|P| = t) and t-order equal to d.Two sets P and P’ are said to be equipotent if |P| = |P′|.Theorem 2. Let m ∈ N be a natural number. Then there exists m nonequipo-tent sets P (|P| = t), over which we can form a U-k – seminet N = (P,B, Π) ofmaximal degree µ = t− d + 2, with t-order equal to d, so that m = t− d.Proof. Let m ∈ N. Then by Theorem1 we can form a U-k – seminet N =(P,B, Π) over the set P, (|P| = t) of maximal degree µ = t− d + 2 for2 ≤ d < t + 22.The maximal number of U-k – seminets 23If in (1) we put m = t− d, i.e. d = t−m, we obtain the following:2 ≤ t−m < t + 22,which finally givesm + 2 ≤ t < 2m + 2. (3)2These inequalities for the variable t have m integer solutions. It means thatthere exist m nonequipotent sets P such that |P| = t, which satisfy the Theorem(see Table 1).24 R. GalićTheorem 3. Let N = (P,B, Π) be a U-k – seminet of maximal degree µ overthe set P, with t-order equal to d. Then for |P| = t ≥ 3 the following holds:a)t + 22< µ ≤ t, (4)b) µ ≤ t < 2µ− 2. (5)Proof. a) Since from (2) it follows that d = t−µ+2, then by substituting into(1) we obtain2 ≤ t− µ + 2 < t + 22.Therefore, because 2 ≤ t− µ + 2, it follows thatµ ≤ t (6)By using the inequality t− µ + 2 < t+22 we gett + 22< µ. (7)From (6) and (7) it directly follows (4) (see Table 2.)b) From (7) it directly follows thatt < 2µ− 2. (8)(6) and (8) now give (5) (see Table 3). 2The results of Theorem1, 2 and 3 corresponding to the maximal degree of a U-k– seminet (over the given set P, (|P| = t) and the given t-order equal to d) is shownin the following tables.References[1] J. Aczel, Quasigroups, nets and nomograms, Advances in Math. 1(1965), 383–450.[2] V. D. Belousov, Algebraičeski seti i kvazigrupp, Štimca, Kǐsnev, 1971.[3] R. Galić, H-k-seminets, Rad JAZU, sv.444(8) (1989), 13–26.[4] R. Galić, U-k – seminets of maximal degree, Glasnik Matematički25(45)(1990), 9–20.[5] V. Havel, Nets and groupoids, Comment. Math. Univ. Carolinne 8(1967),435–449.[6] M. A. Taylor, Classical, cartesian and solution nets, Mathematica13(36)(1971), 151–166.[7] J. Ušan, k-seminets, Mathem. bilten, Skopje, Kn. 1(XXVII)(1977), 41–46.The maximal number of U-k – seminets 25Table 1. Maximal degree µ of a U-k – seminet which depends on the cardinalnumber t of P and t-order equal to d, (resp. m = t− d), that is µ = f(t,m).mt 1 2 3 4 5 6 7 8 9 10 11 12 13 14 153 34 45 4 56 5 67 5 6 78 6 7 89 6 7 8 910 7 8 9 1011 7 8 9 10 1112 8 9 10 11 1213 8 9 10 11 12 1314 9 10 11 12 13 1415 9 10 11 12 13 14 1516 10 11 12 13 14 15 1617 10 11 12 13 14 15 16 1718 11 12 13 14 15 16 1719 11 12 13 14 15 16 1720 12 13 14 15 16 1721 12 13 14 15 16 1722 13 14 15 16 1723 13 14 15 16 1724 14 15 16 1725 14 15 16 1726 15 16 1727 15 16 1728 16 1729 16 1730 1731 17Table 2. Maximal degree µ of a U-k – seminet for the given setP (|P| = t) and t-order equal to d, i.e. µ = f(t, d).t 3 4 5 6 7 8 9 10[t+22]2 3 3 4 4 5 5 6t+22 < µ ≤ t 3 4 4,5 5,6 5,6,7 6,7,8 6,7,8,9 7,8,9,10Table 3. The cardinal number t of the set P for the given maximaldegree µ of a U-k – seminet of the t-order equal to d, i.e. t = f(µ, d).µ 3 4 5 6 7 82µ− 2 4 6 8 10 12 14µ ≤ t < 2µ− 2 3 4,5 5,6,7 6,7,8,9 7,8,9,10,11 8,9,10,11,12,13

English

The maximal number of U-k - seminets of the maximal degree

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Mathematical Communications 2(1997), 21–25 21The maximal number of U-k – seminetsof the maximal degree∗Radoslav Galic´†Abstract. Aczel (1965) investigated quasigroups, 3-nets and nomo-grams and Belousov (1971) k-nets and associated (k-1) – quasigroups.There are different 3 – seminets and k-seminets (see e.g. Havel (1967),Taylor (1971), Usˇan (1977), Galic´ (1989), etc.) to which by some rulesone can assign corresponding algebraic structures (partial quasigroupsand partial groupoids). Galic´ (1990) defines U-k – seminets of the max-imal degree and shows the existence and construction in dependence onthe set P over which one constructs a k-seminet. In this paper it isshown how many U-k – seminets of maximal degree µ can be constructedover the set P for the given t-order.Key words: U-k – seminets, k – seminets, t – order, maximaldegreeSazˇetak. Maksimalni broj U-k – polumrezˇa maksimalnogstupnja. Aczel (1965) proucˇava kvazigrupe, 3 – mrezˇe i nomograme, aBelousov (1971) k – mrezˇe i pridruzˇene (k-1) – kvazigrupe. Postoje ra-zlicˇite 3 – polumrezˇe i k – polumrezˇe (vidi primjerice Havel (1967), Tay-lor (1971), Usˇan (1977), Galic´ (1989) i drugi) kojima se po nekom prav-ilu mogu pridruzˇiti odgovarajuc´e algebarske strukture (parcijalne kvazi-grupe i parcijalni grupoidi). Galic´ (1990) definira U-k – polumrezˇemaksimalnog stupnja te pokazuje postojanje i konstrukciju u ovisnosti oskupu P nad kojim se konstruira k – polumrezˇa. U ovom radu pokazanoje koliko se U-k – polumrezˇa maksimalnog stupnja µ mozˇe konstruiratinad skupom P za zadani t-red.Kljucˇne rijecˇi: U-k – polumrezˇe, k – polumrezˇe, t – red, maksi-malni stupanjAMS subject classifications: 51E30, 20N5Received December 23, 1996, Accepted March 4, 1997∗The lecture presented at the Mathematical Colloquium in Osijek organized by the CroatianMathematical Society – Division Osijek, April 26, 1996.†Faculty of Electrical Engineering, Department of Mathematics, Istarska 3, HR-31 000 Osijek,Croatia, e-mail: galic@drava.etfos.hr22 R. Galic´Let P be a nonempty set and B family of nonempty subsets of P, i.e. B ⊂P(P)\{∅}. Let Π = {X1, ..., Xk}, k ≥ 3, be a partition of the set B. Elements ofthe set P, B and Π are called points, blocks and classes, respectively. Then theordered triple N = (P,B,Π) is called a U-k – seminet if the following conditionsare satisfied:C1. Any two blocks from different classes have at most one point in common, thatis(∀xi ∈ Xi)(∀xj ∈ Xj)(i 6= j ⇒ |xi ∩ xj | ≤ 1),C2. Every point of P lies in exactly one block from each class, that is(∀p ∈ P)(∀i ∈ {1, 2, ..., k})(∃!xi ∈ Xi : p ∈ xi).If in a U-k – seminet N = (P,B,Π) we need to specify classes, then we willdenote this U-k – seminet by N = (P, X1, ..., Xk), where the number k denotes thedegree of the seminet.Number of points of a set A, i.e. the cardinal number of the set A will bedenoted by |A|.Number of points of a maximal block, i.e. the number:d = max{|b| : b ∈ B}is called t-order of a U-k – seminet.N = (P,B,Π) is said to be a U-µ – seminet of maximal degree µ over theset P if for every U-k – seminet N ′ = (P,B′,Π′) over the set P, one has µ ≥ k.The existence and the construction of a U-k – seminet with maximal degree forsome sets P are proved by Galic´ (1990). The following theorem considers that issue:Theorem 1. Let P be a set of points such that |P| = t ≥ 3. Then for2 ≤ d < t+ 22(1)there exists a U -µ – seminet N = (P,B,Π) of maximal degree µ and t-order equalto d over the set P, so thatµ = t− d+ 2. (2)We will consider here maximal degree µ of a U-k – seminet which depends onthe number of points of the set P (|P| = t) and t-order equal to d.Two sets P and P’ are said to be equipotent if |P| = |P′|.Theorem 2. Let m ∈ N be a natural number. Then there exists m nonequipo-tent sets P (|P| = t), over which we can form a U-k – seminet N = (P,B,Π) ofmaximal degree µ = t− d+ 2, with t-order equal to d, so that m = t− d.Proof. Let m ∈ N. Then by Theorem1 we can form a U-k – seminet N =(P,B,Π) over the set P, (|P| = t) of maximal degree µ = t− d+ 2 for2 ≤ d < t+ 22.The maximal number of U-k – seminets 23If in (1) we put m = t− d, i.e. d = t−m, we obtain the following:2 ≤ t−m < t+ 22,which finally givesm+ 2 ≤ t < 2m+ 2. (3)2These inequalities for the variable t have m integer solutions. It means thatthere exist m nonequipotent sets P such that |P| = t, which satisfy the Theorem(see Table 1).24 R. Galic´Theorem 3. Let N = (P,B,Π) be a U-k – seminet of maximal degree µ overthe set P, with t-order equal to d. Then for |P| = t ≥ 3 the following holds:a)t+ 22< µ ≤ t, (4)b) µ ≤ t < 2µ− 2. (5)Proof. a) Since from (2) it follows that d = t−µ+2, then by substituting into(1) we obtain2 ≤ t− µ+ 2 < t+ 22.Therefore, because 2 ≤ t− µ+ 2, it follows thatµ ≤ t (6)By using the inequality t− µ+ 2 < t+22 we gett+ 22< µ. (7)From (6) and (7) it directly follows (4) (see Table 2.)b) From (7) it directly follows thatt < 2µ− 2. (8)(6) and (8) now give (5) (see Table 3). 2The results of Theorem1, 2 and 3 corresponding to the maximal degree of a U-k– seminet (over the given set P, (|P| = t) and the given t-order equal to d) is shownin the following tables.References[1] J. Aczel, Quasigroups, nets and nomograms, Advances in Math. 1(1965), 383–450.[2] V. D. Belousov, Algebraicˇeski seti i kvazigrupp, Sˇtimca, Kiˇsnev, 1971.[3] R. Galic´, H-k-seminets, Rad JAZU, sv.444(8) (1989), 13–26.[4] R. Galic´, U-k – seminets of maximal degree, Glasnik Matematicˇki25(45)(1990), 9–20.[5] V. Havel, Nets and groupoids, Comment. Math. Univ. Carolinne 8(1967),435–449.[6] M. A. Taylor, Classical, cartesian and solution nets, Mathematica13(36)(1971), 151–166.[7] J. Usˇan, k-seminets, Mathem. bilten, Skopje, Kn. 1(XXVII)(1977), 41–46.The maximal number of U-k – seminets 25Table 1. Maximal degree µ of a U-k – seminet which depends on the cardinalnumber t of P and t-order equal to d, (resp. m = t− d), that is µ = f(t,m).mt 1 2 3 4 5 6 7 8 9 10 11 12 13 14 153 34 45 4 56 5 67 5 6 78 6 7 89 6 7 8 910 7 8 9 1011 7 8 9 10 1112 8 9 10 11 1213 8 9 10 11 12 1314 9 10 11 12 13 1415 9 10 11 12 13 14 1516 10 11 12 13 14 15 1617 10 11 12 13 14 15 16 1718 11 12 13 14 15 16 1719 11 12 13 14 15 16 1720 12 13 14 15 16 1721 12 13 14 15 16 1722 13 14 15 16 1723 13 14 15 16 1724 14 15 16 1725 14 15 16 1726 15 16 1727 15 16 1728 16 1729 16 1730 1731 17Table 2. Maximal degree µ of a U-k – seminet for the given setP (|P| = t) and t-order equal to d, i.e. µ = f(t, d).t 3 4 5 6 7 8 9 10[t+22]2 3 3 4 4 5 5 6t+22 < µ ≤ t 3 4 4,5 5,6 5,6,7 6,7,8 6,7,8,9 7,8,9,10Table 3. The cardinal number t of the set P for the given maximaldegree µ of a U-k – seminet of the t-order equal to d, i.e. t = f(µ, d).µ 3 4 5 6 7 82µ− 2 4 6 8 10 12 14µ ≤ t < 2µ− 2 3 4,5 5,6,7 6,7,8,9 7,8,9,10,11 8,9,10,11,12,13

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