Discussiones Mathematicae Algebra and Stochastic Methods 19(1) (1999) 9-40
Jonathan D.H. Smith
Department of Mathematics, Iowa State University
Ames, Iowa 50011, USA
e-mail: jdhsmith@math.iastate.edu
The paper gives a self-contained introduction to modes (idempotent entropic algebras) and modals (modes distributive over semilattices). Constructions of modals from modes are discussed, including a new contravariant submode functor to a category of concave functions. Some characteristic applications of modal theory are presented, including a new identification of the modal structure of multiplication tables of central quasigroups.
Keywords and phrases: mode, modal, entropic, semilattice, affine space, barycentric algebra, algebraic lattice, convex set, concave function, differential group, support function, isotopy, quasigroup.
1991 Mathematics Subject Classification: Primary: 08A99; Secondary: 06A12, 20N05, 51A25, 52A01.
| [1] | F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, NY, 1974. |
| [2] | G. Birkhoff, On the structure of abstract algebras, Proc. Combridge Philos. Soc. 31 (1935), 433-454. |
| [3] | G. Birkhoff, Lattice Theory, (3rd. ed.), Amer. Math. Soc., Providence, RI, 1967. |
| [4] | T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Chelsea, New York, NY, 1948. |
| [5] | K. Borsuk, Multidimensional Analytic Geometry, PWN-Polish Sci. Publ., Warsaw 1969. |
| [6] | R.H. Bruck, What is a loop?, pp. 59-99 in: Studies in Modern Algebra, Prentice-Hall, Englewood Cliffs, NJ, 1963. |
| [7] | B. Csákány, Varieties of affine modules, Acta Sci. Math. 37 (1975), 3-10. |
| [8] | B.A. Davey and G. Davis, Tensor products and entropic varieties, Algebra Universalis 21 (1985), 68-88. |
| [9] | J. Flood, Semiconvex geometry, J. Austral. Math. Soc. 30 (1981), 496-510. |
| [10] | G. Gierz, et al., A Compendium of Continuous Lattices, Springer-Verlag, Berlin 1980. |
| [11] | J.S. Golan, The Theory of Semirings, Longman, Harlow 1992. |
| [12] | P.A. Grillet, The tensor product of semigroups, Trans. Amer. Math. Soc. 138 (1969), 267-180. |
| [13] | K.H. Hofmann, M. Mislove and A. Stralka, The Pontryagin Duality of Compact 0-Dimensional Semilattices and its Applications, Springer-Verlag, Berlin 1974. |
| [14] | P.T. Johnstone, Stone Spaces, Cambridge Univ. Press, Cambridge 1982. |
| [15] | K.A. Kearnes, Semilattice modes I: the associated semiring, Algebra Universalis 34 (1995), 220-272. |
| [16] | S. Mac Lane, Homology, Springer-Verlag, Berlin 1963. |
| [17] | S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag, Berlin 1971. |
| [18] | B. Mielnik, Theory of filters, Comm. Math. Phys. 15 (1969), 1-46. |
| [19] | W.D. Neumann, On the quasivariety of convex subsets of affine spaces, Arch. Math. 21 (1970), 11-16. |
| [20] | F. Ostermann and J. Schmidt, Baryzentrischer Kalkül als axiomatische Grundlage der affinen Geometrie, J. reine angew. Math. 224 (1966), 44-57. |
| [21] | A. Pilitowska, A. Romanowska and J.D.H. Smith, Affine spaces and algebras of subalgebras, Algebra Universalis 34 (1995), 527-540. |
| [22] | A. Romanowska, An introduction to the theory of modes and modals, Contemp. Math. 131 (1992), 241-262. |
| [23] | A.B. Romanowska and J.D.H. Smith, Bisemilattices of subsemilattices, J. Algebra 70 (1981), 78-88. |
| [24] | -, Modal Theory, Heldermann-Verlag, Berlin 1985. |
| [25] | -, From affine to projective geometry via convexity, pp. 255-269 in: ``Universal Algebra and Lattice Theory", Springer-Verlag, Berlin 1985. |
| [26] | -, Support functions and ordinal products, Geom. Dedicata 30 (1989), 281-296. |
| [27] | -, Communicating processes and entropic algebras, Fund. Inform. 13 (1990), 263-274. |
| [28] | -, On the structure of barycentric algebras, Houston J. Math. 16 (1990), 431-447. |
| [29] | -, On the structure of semilattice sums, Czechoslovak Math. J. 41 (1991), 24-43. |
| [30] | -, Differential groupoids, Contributions to General Algebra 7 (1991), 283-290. |
| [31] | -, Separable modes, Algebra Universalis 30 (1993), 61-71. |
| [32] | -, Duality for semilattice representations, J. Pure Appl. Algebra 115 (1997), 289-308. |
| [33] | E. Slatinský, Die arithmetische Operation der Summe, Archiv. Math. (Brno) 20 (1984), 9-20. |
|
J.D.H. Smith, Finite equationally complete entropic quasigroups, Contributions to General Algebra 1 (1979), 345-356. |
| [35] | -, Centrality, pp. 95-114 in: Quasigroups and Loops: Theory and Applications, Helderman-Verlag, Berlin 1990. |
| [36] | -, Entropy, character theory and centrality of finite quasigroups, Math. Proc. Cambridge Philos. Soc. 108 (1990), 435-443. |
| [37] | -, Homotopy and semisymmetry of quasigroups, Algebra Universalis 38 (1997), 175-184. |
| [38] | Á. Szendrei, On the arity of affine modules, Colloq. Math. 38 (1977), 1-4. |
Received 27 November 1997
Revised 25 February 1998