Topological characterizations of ordered groups with quasi-divisor theory. (English) Zbl 1019.06008

Močkoř, Jiří

Czech. Math. J. 52, No. 3, 595-607 (2002).

Summary: For an order embedding \(G @>h>>\Gamma \) of a partially ordered group \(G\) into an \(l\)-group \(\Gamma \) a topology \(\mathcal T_{\widehat W}\) is introduced on \(\Gamma\) that is defined by a family of valuations \(W\) on \(G\). Some density properties of the sets \(h(G)\), \(h(X_t)\) and \((h(X_t)\setminus \{h(g_1),\dots ,h(g_n)\})\) (\(X_t\) being the \(t\)-ideals in \(G\)) in the topological space \((\Gamma ,\mathcal T_{\widehat W})\) are then investigated, each of them being equivalent to the statement that \(h\)  is a strong theory of quasi-divisors.

MSC:

06F15

Ordered groups

Keywords:

quasi-divisor theory; ordered group; valuations; \(t\)-ideal

References:

[1]	I. Arnold: Ideale in kommutativen Halbgruppen. Rec. Math. Soc. Math. Moscow 36 (1929), 401-407. · JFM 55.0681.03
[2]	M. Anderson and T. Feil: Lattice-ordered Groups. D. Reidl Publ. Co., Dordrecht, Tokyo, 1988. · Zbl 0636.06008
[3]	K. E. Aubert: Divisors of finite character. Ann. Mat. Pura Appl. 33 (1983), 327-361. · Zbl 0533.20034 · doi:10.1007/BF01766024
[4]	K. E. Aubert: Localizations dans les systémes d’idéaux. C. R. Acad. Sci. Paris 272 (1971), 465-468. · Zbl 0216.05001
[5]	Z. I. Borevich and I. R. Shafarevich: Number Theory. Academic Press, New York, 1966. · Zbl 0145.04902
[6]	P. Conrad: Lattice Ordered Groups. Tulane University, 1970. · Zbl 0258.06011
[7]	L. G. Chouinard: Krull semigroups and divisor class group. Canad. J. Math. 33 (1981), 1459-1468. · Zbl 0453.13005 · doi:10.4153/CJM-1981-112-x
[8]	A. Geroldinger and J. Močkoř: Quasi-divisor theories and generalizations of Krull domains. J. Pure Appl. Algebra 102 (1995), 289-311. · Zbl 0853.13012 · doi:10.1016/0022-4049(94)00088-Z
[9]	R. Gilmer: Multiplicative Ideal Theory. M. Dekker, Inc., New York, 1972. · Zbl 0248.13001
[10]	M. Griffin: Rings of Krull type. J. Reine Angew. Math. 229 (1968), 1-27. · Zbl 0173.03504 · doi:10.1515/crll.1968.229.1
[11]	M. Griffin: Some results on \(v\)-multiplication rings. Canad. J. Math. 19 (1967), 710-722. · Zbl 0148.26701 · doi:10.4153/CJM-1967-065-8
[12]	P. Jaffard: Les systémes d’idéaux. Dunod, Paris, 1960. · Zbl 0101.27502
[13]	J. Močkoř: Groups of Divisibility. D. Reidl Publ. Co., Dordrecht, 1983. · Zbl 0528.13001
[14]	J. Močkoř and J. Alajbegovic: Approximation Theorems in Commutative Algebra. Kluwer Academic publ., Dordrecht, 1992.
[15]	J. Močkoř and A. Kontolatou: Groups with quasi-divisor theory. Comm. Math. Univ. St. Pauli, Tokyo 42 (1993), 23-36. · Zbl 0794.06015
[16]	J. Močkoř and A. Kontolatou: Divisor class groups of ordered subgroups. Acta Math. Inform. Univ. Ostraviensis 1 (1993), 37-46. · Zbl 0849.06013
[17]	J. Močkoř and A. Kontolatou: Quasi-divisors theory of partly ordered groups. Grazer Math. Ber. 318 (1992), 81-98. · Zbl 0794.06014
[18]	J. Močkoř: \(t\)-valuation and theory of quasi-divisors. J. Pure Appl. Algebra 120 (1997), 51-65. · Zbl 0885.06008 · doi:10.1016/S0022-4049(96)00059-X
[19]	J. Močkoř and A. Kontolatou: Some remarks on Lorezen \(r\)-group of partly ordered group. Czechoslovak Math. J. 46(121) (1996), 537-552. · Zbl 0879.20031
[20]	J. Močkoř: Divisor class group and the theory of quasi-divisors. To appear. · Zbl 0955.06011
[21]	J. Ohm: Semi-valuations and groups of divisibility. Canad. J. Math. 21 (1969), 576-591. · Zbl 0177.06501 · doi:10.4153/CJM-1969-065-9
[22]	L. Skula: Divisorentheorie einer Halbgruppe. Math. Z. 114 (1970), 113-120. · Zbl 0177.03202 · doi:10.1007/BF01110320
[23]	L. Skula: On \(c\)-semigroups. Acta Arith. 31 (1976), 247-257. · Zbl 0303.13014