Independence with respect to family of mappings in abstract algebras

Głazek, Kazimierz

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Independence with respect to family of mappings in abstract algebras

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Kazimierz Głazek

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Rozprawy Matematyczne tom/nr w serii: 81 wydano: 1971

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CONTENTS
INTRODUCTION .................................................................................................................................................................................................. 5
I. INDEPENDENCE WITH RESPECT TO A GIVEN FAMILY OF MAPPINGS (GENERAL PROPERTIES) ............................................ 7
§ 1. Notation and main definitions.................................................................................................................................................................... 7
§ 2. Notions of independence defined by families of mappings (Q-independence).............................................................................. 9
§ 3. Maximal families of mappings for a given independence.................................................................................................................... 13
§ 4. Q-independent sets of generators (Q-bases)......................................................................................................................................... 17
§ 5. Exchange of Q-independent sets.............................................................................................................................................................. 27
II. VARIOUS NOTIONS OF INDEPENDENCE IN ALGEBRAS AND LINEAR SPACES............................................................................. 29
§ 6. Construction of some family of mappings .............................................................................................................................................. 29
§ 7. Corollaries concerning v**-algebras and linear spaces....................................................................................................................... 31
III. THE INDEPENDENCE NOTIONS IN ABELIAN GROUPS AND QUASI-LINEAR ALGEBRAS............................................................ 33
§ 8. $S_0$- and S-independence in abelian groups.................................................................................................................................... 33
§ 9. The S-, $S_0-$, G-, and R-independence in quasi-linear algebras................................................................................................... 37
IV. VARIOUS NOTIONS OF INDEPENDENCE IN BOOLEAN ALGEBRAS AND SOME OF THEIR REDUCTS.................................... 46
§ 10. Additional notations, and some known results.................................................................................................................................... 45
§ 11. Various notions of independence in regular reducts of Boolean algebra....................................................................................... 47
REFERENCES...................................................................................................................................................................................................... 54

Słowa kluczowe

Tematy

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Seria

Rozprawy Matematyczne tom/nr w serii: 81

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Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae, Tom LXXXI

Daty

wydano

1971

Twórcy

autor

Kazimierz Głazek

Institute of Mathematics of the Wrocław University

Bibliografia

[1] G. Birkhoff, Latice Theory, New York 1948.
[2] S. Fajtlowicz and K. Głazek, Independence in separable variables algebras, Colloq. Math. 17 (1967), pp. 221-224.
[3] S. Fajtlowicz, K. Głazek and. K. Urbanik, Separable variables algebras Colloq. Math. 16 (1986), pp. 161-171.
[4] B. Gleichgewicht and K. Głazek, Remarks on n-groups as abstract algebras, Colloq. Math. 17 (1967), pp. 209-219.
{5] K. Głazek, On weak automorphisms of quasi-linear algebras, Colloq. Math., in print.
[6] A. Goetz and. C. Ryll-Nardzewski, On bases of abstract algebras, Bull. Aoad, Polon. Sci. 8 (1960), pp. 157-161.
[7] G. Graetzer, Universal algebra, mimeographed, The Pensylvania State University, 1964.
[8] G. Graetzer, A new notion of independence in universal algebra, Colloq. Math. 17 (1967), pp. 225-234.
[9] A. Hulanicki, E. Marczewski and J. Mycielski, Exchange of independent sets in abstract algebras I, Colloq. Math. 14 (1966), pp. 203-215.
[10] Б. И. Плоткин, Группы автоморфизмов алгебраических систем, Москва 1966.
[11] E. Marczewski, A general scheme of the notions of independence in mathematics, Bull. Acad. Polon. Sci., 6 (1958) pp. 731-736.
[12] E. Marczewski, Independence in algebras of sets and Boolean algebras, Fund. Math. 48 (1960) pp. 135-145.
[13] E. Marczewski, Independence and homomorphisms in abstract algebras, Fund. Math. 50 (1961), pp. 45-61.
[14] E. Marczewski, Homogeneous operations and homogeneous algebras, Fund. Math., 56 (1964), pp. 81-103.
[15] E. Marczewski, Fermeture generalisée et notions d'independence, Celebrazioni archimedes de secolo XX, Simposio di topologia, 1964, pp. 21-32.
[16] E. Marczewski, Independence in abstract algebras, Results and problems, Colloq. Math., 14(1966), pp. 169-188.
[17] E. Marczewski, Independence with respect lo a family of mappings, Colloq. Math., 20 (1968), pp. 11-17.
[18] W. Markiewicz, Independence in a certain class of abstract algebras, Fund. Math., 50 (1961), pp. 333-340.
[19] W. Markiewicz, On a certain class of abstract algebras, Fund. Math., 54 (1964), p. 115-124.
[20] J. Schmidt, Eine algebraische Äquivalente zum Auswahlaxiom,, Fund. Math., 50 (1962), pp. 485-496.
[21] S. Świerczkowski, Topologies in free algebras, Proc. London Math. Soc., (3) 14 (1964), pp. 566-576.
[22] S. Świerczkowski, On independent elements infinitely generated algebras, Bull. Acad. Polon, Sci., 6 (1958), pp. 749-752.
[23] K. Urbanik, A representation theorem for Marczewski's algebras, Fund. Math., 48 (1960), pp. 147-167.
[24] K. Urbanik, Linear independence in abstract algebras, Colloq. Math., 15 (1966), pp. 233-255.

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