Undecidability of representability as binary relations

In this article we establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for Jonsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions.

[1]  Hajnal Andréka,et al.  Representations of distributive lattice-ordered semigroups with binary relations , 1991 .

[2]  Bjarni Jónsson,et al.  Representation of modular lattices and of relation algebras , 1959 .

[3]  Dexter Kozen,et al.  On the Complexity of the Horn Theory of REL , 2003 .

[4]  Robin Hirsch.  The Class of Representable Ordered Monoids has a Recursively Enumerable, Universal Axiomatisation but it is Not Finitely Axiomatisable , 2005, Log. J. IGPL.

[5]  M. Stone.  The theory of representations for Boolean algebras , 1936 .

[6]  H. Andréka,et al.  The equational theory of union-free algebras of relations , 1995 .

[7]  Y. Venema.  Atom structures and Sahlqvist equations , 1997 .

[8]  Boris M. Schein,et al.  Relation algebras and function semigroups , 1970 .

[9]  Marcel Jackson.  Some undecidable embedding problems for finite semigroups , 1999 .

[10]  I. Németi,et al.  Decidability of relation algebras with weakened associativity , 1987 .

[11]  R. Lyndon.  THE REPRESENTATION OF RELATIONAL ALGEBRAS , 1950 .

[12]  Marcel Jackson,et al.  An Invitation to C-semigroups , 2001 .

[13]  A. Tarski,et al.  Boolean Algebras with Operators. Part I , 1951 .

[14]  Marcel Jackson,et al.  Modal restriction Semigroups: towards an Algebra of Functions , 2011, Int. J. Algebra Comput..

[15]  Mark Kambites,et al.  Faithful Functors from Cancellative Categories to Cancellative Monoids with an Application to Abundant Semigroups , 2005, Int. J. Algebra Comput..

[16]  Mikhail V. Volkov,et al.  Relatively inherently nonfinitely q-based semigroups , 2008 .

[17]  J. Howie.  Fundamentals of semigroup theory , 1995 .

[18]  B. M. Schein,et al.  Representations of ordered semigroups and lattices by binary relations , 1978 .

[19]  A. Tarski,et al.  Boolean Algebras with Operators , 1952 .

[20]  Szabolcs Mikulás,et al.  Representable semilattice-ordered monoids , 2007 .

[21]  M. Jackson,et al.  Undecidable problems for completely 0-simple semigroups , 2009 .

[22]  Robin Hirsch,et al.  Representability is not decidable for finite relation algebras , 1999 .

[23]  John Fountain.  Free right type A semigroups , 1991 .

[24]  Georg Struth,et al.  Algebras of modal operators and partial correctness , 2006, Theor. Comput. Sci..

[25]  R. Lyndon.  THE REPRESENTATION OF RELATION ALGEBRAS, II , 1956 .

[26]  D. Monk.  On representable relation algebras. , 1964 .

[27]  Vaughan R. Pratt,et al.  Dynamic algebras as a well-behaved fragment of relation algebras , 1988, Algebraic Logic and Universal Algebra in Computer Science.

[28]  Mark V. Sapir,et al.  Algorithmic Problems in Varieties , 1995, Int. J. Algebra Comput..

[29]  G. Higman,et al.  A Finitely Generated Infinite Simple Group , 1951 .

[30]  R. Robinson.  Undecidability and nonperiodicity for tilings of the plane , 1971 .

[31]  Boris M. Schein.  Subsemigroups of inverse semigroups , 1997 .

[32]  A. M. Slobodskoi.  Unsolvability of the universal theory of finite groups , 1981 .

[33]  Marcel Jackson,et al.  Partial Maps with Domain and Range: Extending Schein's Representation , 2009 .

[34]  S. Margolis,et al.  Algorithmic problems for finite groups and finite 0-simple semigroups , 1997 .

[35]  Georg Struth,et al.  Kleene algebra with domain , 2003, TOCL.

[36]  Yde Venema,et al.  Canonical varieties with no canonical axiomatisation , 2004 .

[37]  Stephen D. Comer,et al.  A remark on representable positive cylindric algebras , 1991 .

[38]  D. A. Bredikhin,et al.  Reducts of Tarski relation algebras , 1998 .

[39]  I. Hodkinson,et al.  Relation Algebras by Games , 2002 .

[40]  Szabolcs Mikulás,et al.  Axiomatizability of reducts of algebras of relations , 2000 .

[41]  Trevor Evans,et al.  Embeddability and the Word Problem , 1953 .