A survey of clones on infinite sets

Abstract.We summarize what we know about the clone lattice on an infinite set and formulate what we consider the most important open problems.

[1]  Robert W. Quackenbush.  A survey of minimal clones , 1995 .

[2]  Y. Moschovakis.  Descriptive Set Theory , 1980 .

[3]  Large intervals in the clone lattice , 2002, math/0208066.

[4]  Analytic clones , 2004, math/0404214.

[5]  A Survey of Clones Closed Under Conjugation , 2004 .

[6]  Precomplete classes of operations on an uncountable set , 1985 .

[7]  Ágnes Szendrei,et al.  Clones in universal algebra , 1986 .

[8]  Wacław Sierpiński,et al.  Sur les fonctions de plusieurs variables , 1945 .

[9]  S. Katsman,et al.  COMMUTATIVE SEMIGROUPS WHOSE LATTICE OF SUBSEMIGROUPS SATISFIES A NONTRIVIAL IDENTITY , 1990 .

[10]  L. Heindorf.  The maximal clones on countable sets that include all permutations , 2002 .

[11]  The Clone Space as a Metric Space , 1998 .

[12]  Emil L. Post.  The two-valued iterative systems of mathematical logic , 1942 .

[13]  Identities in the lattices of closed classes , 1993 .

[15]  Ivo G. Rosenberg,et al.  Locally Maximal Clones II , 2000, J. Autom. Lang. Comb..

[16]  Some maximal closed classes of operations on infinite sets , 1974 .

[17]  Conditions satisfied by clone lattices , 2001 .

[18]  Ivo G. Rosenberg,et al.  Locally Maximal Clones , 1982, J. Inf. Process. Cybern..

[19]  A. Kechris.  Classical descriptive set theory , 1987 .

[20]  Основные отношения $S$-классификации функций многозначной логики@@@Basic relations for the $S$-classification of functions of multivalued logic , 1996 .

[21]  R. P. Dilworth,et al.  Algebraic theory of lattices , 1973 .

[22]  Ivo G. Rosenberg,et al.  A classification of universal algebras by infinitary relations , 1971 .

[23]  GÁBOR CZÉDLI,et al.  A CLASS OF CLONES ON COUNTABLE SETS ARISING FROM IDEALS , 2001 .

[24]  I. Rosenberg.  MINIMAL CLONES I: THE FIVE TYPES , 1986 .

[25]  D. Lau,et al.  Function Algebras on Finite Sets: Basic Course on Many-Valued Logic and Clone Theory (Springer Monographs in Mathematics) , 2006 .

[26]  L. A. Kaluzhnin,et al.  Galois theory for post algebras. I , 1969 .

[27]  I. G. Rosenberg,et al.  Local completeness I , 1984 .

[28]  Clones containing all almost unary functions , 2004, math/0401102.

[29]  M. Pinsker.  Precomplete Clones on Infinite Sets which are Closed under Conjugation , 2004, math/0409217.

[30]  B. A. Romov.  Galois correspondence between iterative post algebras and relations on an infinite set , 1977 .

[31]  Reinhard Pöschel,et al.  Funktionen- und Relationenalgebren , 1979 .

[32]  W. Wistar Comfort,et al.  The Theory of Ultrafilters , 1974 .

[33]  Michael Pinsker.  Monoidal intervals of clones on infinite sets , 2008, Discret. Math..

[34]  Finite sublattices in the lattice of clones , 1994 .

[35]  D. Lau,et al.  Function algebras on finite sets : a basic course on many-valued logic and clone theory , 2006 .

[36]  M. Pinsker.  Maximal clones on uncountable sets that include all permutations , 2004, math/0401103.

[37]  B. Csákány.  Minimal clones—a minicourse , 2005 .

[38]  M. Pinsker.  THE NUMBER OF UNARY CLONES CONTAINING THE PERMUTATIONS ON AN INFINITE SET , 2004, math/0410406.

[39]  The Minimal Clones above the Permutations , 2005, math/0512367.

[40]  J. Whitfield.  All creatures great and small , 2001, Nature.

[41]  L. A. Kaluzhnin,et al.  Galois theory for Post algebras. II , 1969 .

[42]  Saharon Shelah,et al.  Clones on regular cardinals , 2000 .

[43]  Michael Pinsker.  Algebraic lattices are complete sublattices of the clone lattice over an infinite set , 2006 .

[44]  F. R. Drake,et al.  COMBINATORIAL SET THEORY: PARTITION RELATIONS FOR CARDINALS (Studies in Logic and the Foundations of Mathematics, 106) , 1986 .

[45]  Jerzy Słupecki,et al.  A criterion of fullness of many-valued systems of propositional logic , 1972 .