Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł książki

## Distributive multisemilattices

Autorzy
Seria
Rozprawy Matematyczne tom/nr w serii: 309 wydano: 1991
Zawartość
Warianty tytułu
Abstrakty
EN
A distributive multisemilattice of type n is an algebra with a family of n binary semilattice operations on a common carrier that are mutually distributive. This concept for n=2 comprises the distributive bisemilattices (or quasilattices), of which distributive lattices and semilattices with duplicated operations are the best known examples. Multisemilattices need not satisfy the absorption law, which holds in all lattices.
Kalman has exhibited a subdirectly irreducible distributive bisemilattice which is neither a lattice nor a semilattice. It has three elements. In this paper it is shown that all the subdirectly irreducible distributive multisemilattices are derived from those for n=2 simply by duplicating their operations in all possible ways. Thus, up to isomorphism there are $2^{n}-1$ of type n, but up to the coarser relation of polynomial equivalence there are only three. Hence every distributive multisemilattice is the subdirect product of irreducibles, each with two or three elements.
The rest of the paper is devoted to the varieties of distributive multisemilattices. The lattice of these varieties is described, and bases for their identities are given.
EN

CONTENTS
1. Introduction........................................................................................................................5
2. Definition, basic examples and properties of multisemilattices...........................................6
3. The subdirectly irreducibles..............................................................................................13
4. The lattice of subvarieties of $D_n$.................................................................................18
5. Subvarieties of $D_n$ defined by identities involving at most two operation symbols......24
6. Some further comments and open problems....................................................................34
References...........................................................................................................................40
Słowa kluczowe
Tematy
Kategoryzacja MSC:
Miejsce publikacji
Warszawa
Seria
Rozprawy Matematyczne tom/nr w serii: 309
Liczba stron
42
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCIX
Daty
wydano
1991
otrzymano
1990-05-31
Twórcy
autor
• Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003, U.S.A.
autor
• Institute of Mathematics, Warsaw Technical University, 00-661 Warszawa, Poland
Bibliografia
• [A] B. H. Arnold, Distributive lattices with a third operation defined, Pacific J. Math. 1 (1951), 33-41.
• [B] R. Balbes, A representation theorem for distributive quasi-lattices, Fund. Math. 68 (1970), 207-214.
• [BH] H.-J. Bandelt and J. Hedlíková, Median algebras, Discrete Math. 45 (1983), 1-30.
• [Bi] G. Birkhoff, Lattice Theory, 3rd ed., Amer. Math. Soc., Providence, R.I., 1967.
• [Bi1] G. Birkhoff, Subdirect unions in universal algebra, Bull. Amer. Math. Soc. 50 (1944), 764-768.
• [BK] G. Birkhoff and S. A. Kiss, A ternary operation in distributive lattices, ibid. 53 (1947), 749-752.
• [C] P. M. Cohn, Universal Algebra, Reidel, Dordrecht 1981.
• [D1] J. Dudek, On bisemilattices I, Colloq. Math. 47 (1982), 1-5.
• [D2] J. Dudek, On bisemilattices II, to appear.
• [D3] J. Dudek, On bisemilattices III, Math. Sem. Notes Kobe Univ. 10 (1982), 275-279.
• [DG] J. Dudek and E. Graczyńska, The lattice of varieties of algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. 29 (1981), 337-340.
• [DR] J. Dudek and A. Romanowska, Bisemilattices with four essentially binary polynomials, in: Colloq. Math. Soc. J. Bolyai 33, North-Holland, 1983, 337-360.
• [G1] J. Gałuszka, Generalized absorption laws in bisemilattices, Algebra Universalis 19 (1984), 304-318.
• [G2] J. Gałuszka, On bisemilattices with generalized absorption laws, I, Demonstratio Math. 20 (1987), 37-43.
• [G3] J. Gałuszka, Bisemilattices with five essentially binary polynomials, Math. Slovaca 38 (1988), 123-132.
• [GR] G. Gierz and A. Romanowska, Duality for distributive bisemilattices, J. Austral. Math. Soc., to appear.
• [Gi] M. Ginsberg, Bilattices, preprint, Dept. of Computer Science, Stanford University, 1986.
• [Go1] R. Godowski, On bisemilattices with one absorption law, Demonstratio Math. 19 (1986), 304-318.
• [Go2] R. Godowski, On bisemilattices with co-absorption law, preprint.
• [Gr1] G. Grätzer, General Lattice Theory, Akademie-Verlag, Berlin 1978.
• [Gr2] G. Grätzer, Universal Algebra, 2nd ed., Springer, New York 1979.
• [Gra] A. A. Grau, Ternary Operations and Boolean Algebra, Ph.D. Thesis, Univ. of Michigan, 1944.
• [H] T. J. Head, The varieties of commutative monoids, Nieuw Arch. Wisk. 16 (1968), 203-206.
• [JK1] J. Jakubík and M. Kolibiar, On some properties of a pair of lattices, Czechoslovak Math. J. 4 (79) (1954), 1-27 (in Russian).
• [JK2] J. Jakubík and M. Kolibiar, Lattices with a third distributive operation, Math. Slovaca 27 (1977), 287-292.
• [J] B. Jónsson, Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110-121.
• [K] J. A. Kalman, Subdirect decomposition of distributive quasilattices, Fund. Math. 71 (1971), 161-163.
• [Ki] A. Kisielewicz, A solution of Dedekind's problem on the number of isotone Boolean functions, J. Reine Angew. Math. 3 (1988), 1-6.
• [Kn1] A. Knoebel, A comment on Balbes' representation theorem for distributive quasi-lattices, Fund. Math. 90 (1976), 187-188.
• [Kn2] A. Knoebel, The equational classes generated by single functionally precomplete algebras, Mem. Amer. Math. Soc. 57 (332) (1985).
• [Kn3] A. Knoebel, Distributive polylattices, Abstracts Amer. Math. Soc. 7 (1986), 19.
• [LPP] H. Lakser, R. Padmanabhan and C. R. Platt, Subdirect decomposition of Płonka sums, Duke Math. J. 39 (1972), 485-488.
• [M1] A. I. Mal'cev, Algebraic Systems, Springer, Berlin 1973.
• [M2] A. I. Mal'cev, Multiplication of classes of algebraic systems, Sibirsk. Mat. Zh. 8 (1967), 346-365 (in Russian).
• [MMT] R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Wadsworth & Brooks/Cole, Monterey, Cal., 1987.
• [MR] R. McKenzie and A. Romanowska, Varieties of ·-distributive bisemilattices, in: Contributions to General Algebra, Proc. Klagenfurt Conference 1978, Verlag Johannes Heyn, 1978, 213-218.
• [Me] G. C. Meletiou, Linear spaces as distributive k-quasilattices, Demonstratio Math., to appear.
• [N] J. Nieminen, Ideals in distributive quasi-lattices, Studia Univ. Babeş-Bolyai Math. 1977 (1), 6-11.
• [Pa] R. Padmanabhan, Regular identities in lattices, Trans. Amer. Math. Soc. 158 (1971), 179-188.
• [Pn] F. Pastijn, Idempotent distributive semirings II, Semigroup Forum 26 (1983), 151-166.
• [PR] F. Pastijn and A. Romanowska, Idempotent distributive semirings I, Acta Sci. Math. (Szeged) 44 (1982), 239-253.
• [P1] J. Płonka, On distributive quasi-lattices, Fund. Math. 60 (1967), 191-200.
• [P2] J. Płonka, On a method of construction of abstract algebras, ibid. 61 (1967), 183-189.
• [P3] J. Płonka, On distributive n-lattices and n-quasilattices, ibid. 62 (1968), 293-300.
• [P4] J. Płonka, On equational classes of abstract algebras defined by regular equations, ibid. 64 (1969), 241-247.
• [P5] J. Płonka, On free algebras and algebraic decompositions of algebras from some equational classes defined by regular equations, Algebra Universalis 1 (1971), 261-267.
• [R1] A. Romanowska, On free algebras in some equational classes defined by regular equations, Demonstratio Math. 11 (1978), 1131-1137.
• [R2] A. Romanowska, On bisemilattices with one distributive law, Algebra Universalis 10 (1980), 36-47.
• [R3] A. Romanowska, Subdirectly irreducible ·-distributive bisemilattices, I, Demonstratio Math. 13 (1980), 767-785.
• [R4] A. Romanowska, Free bisemilattices with one distributive law, ibid., 565-572.
• [R5] A. Romanowska, Idempotent distributive semirings with a semilattice reduct, Math. Japon. 27 (1982), 483-493.
• [R6] A. Romanowska, Free idempotent distributive semirings with a semilattice reduct, ibid., 467-481.
• [R7] A. Romanowska, On algebras of functions from partially ordered sets into distributive lattices, in: Lecture Notes in Math. 1004, Springer, 1983, 245-256.
• [R8] A. Romanowska, Building bisemilattices from lattices and semilattices, in: Contributions to General Algebra 2, Proc. Klagenfurt Conference 1982, Verlag Hölder-Pichler-Tempsky, 1983, 343-358.
• [R9] A. Romanowska, On some constructions of bisemilattices, Demonstratio Math. 17 (1984), 1011-1021.
• [R10] A. Romanowska, Constructing and reconstructing of algebras, ibid. 18 (1985), 209-230.
• [R11] A. Romanowska, On regular and regularized varieties, Algebra Universalis 23 (1986), 215-241.
• [RS1] A. Romanowska and J. D. H. Smith, Bisemilattices of subsemilattices, J. Algebra 70 (1981), 78-88.
• [RS2] A. Romanowska and J. D. H. Smith, Distributive lattices, generalisations, and related non-associative structures, Houston J. Math. 11 (1985), 367-383.
• [RS3] A. Romanowska and J. D. H. Smith, Model Theory, and Algebraic Approach to Order, Geometry and Convexity, Heldermann, Berlin 1985.
• [RS4] A. Romanowska and J. D. H. Smith, On the structure of semilattice sums, preprint, 1987.
• [RT] A. Romanowska and A. Trakul, On the structure of some bilattices, in: Universal and Applied Algebras, Proc. Universal Algebra Symposium, Turawa 1988, K. Halkowska and B. Stawski (eds.), World Scientific, Singapore 1989, 235-253.
• [S] B. M. Schein, Homomorphisms and subdirect decompositions of semigroups, Pacific J. Math. 17 (1966), 529-547.
• [T] A. Trakul, Bilattices, Master thesis, Warsaw Technical University, 1988 (in Polish).
• [W] R. O. Winder, Threshold Logic, Ph.D. Thesis, Princeton Univ., 1962.
Języki publikacji
 EN
Uwagi
1985 Mathematics Subject Classification: Primary 06A12, 08B15; Secondary 05C40, 08B05.