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2-Semilattices: Residual Properties and Applications to the Constraint Satisfaction Problem

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2-Semilattices: Residual Properties and Applications to the Constraint Satisfaction Problem 2-Semilattices: Residual Properties and Applications to the Constraint Satisfaction Problem by Ian Payne A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Pure Mathematics Waterloo, Ontario, Canada, 2017 c Ian Payne 2017 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner George McNulty Professor University of South Carolina Supervisor Ross Willard Professor University of Waterloo Internal Member Stanley Burris Distinguished Professor Emeritus Adjunct Professor University of Waterloo Internal Member Rahim Moosa Professor University of Waterloo Internal-external Member Chris Godsil Professor University of Waterloo ii I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Abstract Semilattices are algebras known to have an important connection to partially ordered sets. In particular, if a partially ordered set (A; ≤) has greatest lower bounds, a semilattice (A; ^) can be associated to the order where a ^ b is the greatest lower bound of a and b. In this thesis, we study a class of algebras known as 2-semilattices, which is a generalization of the class of semilattices. Similar to the correspondence between partial orders and semilattices, there is a correspondence between certain digraphs and 2-semilattices. That is, to every 2-semilattice, there is an associated digraph which holds information about the 2-semilattice. Making frequent use of this correspondence, we explore the class of 2-semilattices from three perspectives: (i) Tame Congruence Theory, (ii) the \residual character" of the class of 2-semilattices, and (iii), the constraint satisfaction problem. Tame Congruence Theory, developed in [29], is a structure theory on finite algebras driven by understanding their prime congruence quotients. The theory assigns to each such quotient a type from 1 to 5. We show that types 3, 4, and 5 can occur in the class of 2-semilattices, but type 4 can not occur in a finite simple 2-semilattice. Classes of algebras contain \subdirectly irreducible" members which hold information about the class. Specifically, the size of these members has been of interest to many authors. We show for certain subclasses of the class of 2-semilattices that there is no cardinal bound on the size of the irreducible members in that subclass. The “fixed template constraint satisfaction problem" can be identified with the decision problem hom(A) where A is a fixed finite relational structure. The input to hom(A) is a finite structure B similar to A. The question asked is \does there exist a homomorphism from B to A?" Feder and Vardi [22] conjectured that for fixed A, this decision problem is either NP-complete or solvable in polynomial time. Bulatov [15] confirmed this conjecture in the case that A is invariant under a 2-semilattice operation. We extend this result. iv Acknowledgements There are many people I would like to thank for many different things. First of all, thanks to my parents, Rita and Scott for nurturing my love of math from when I was very young. Thanks to my siblings Anthony, Cheyanne, Katie, and Natalie, as well as my unwavering best friends Craig, Ditto, and Ryan. Your love and support made my life in Newfoundland as perfect an upbringing as anyone could hope for. Thanks to my uncles, Kirby and Milton, my aunts Nicol and Glennis, and their collective offspring Adam, Bianca, Ella, Evelyn, and Jeffrey. Because of you, I never really left home. There isn't room on this page to list all of the new friends who made my time in graduate school as enjoyable as it was. In no particular order, thanks to Zack, Pat, Omar, Ale, Jordan, Parisa, Robert, Jess, Seda, Mike (Szestopalow), Matt, Farrah, and Mike (McTavish). Thanks Blake for being a great colleague, roommate, and above all, buddy. Thanks to Michael for the laughs and for sharing and supporting my love of elementary problems. Thanks to Ty for being there from the first day until the last one. The easiest way to clean our office may be with gas and a match. Thanks to Bill for the stories, counseling, and patience. Most of all, thanks to Carrie for tolerating me these last four years. Thanks to Chris, George, Rahim, and Stan who read my thesis and provided valuable feedback. Finally I would like to thank my supervisor, Ross Willard. I have felt lucky to have your advice and support through this whole journey. Thanks for taking me to all the conferences, helping me when I needed it, not helping me when I didn't need it, and somehow always being in a good mood even at times when you must have wanted to slap me. I sincerely hope that our time working together isn't over. v Dedication This is dedicated to my parents, Rita and Scott. Now that you have retired, you should have plenty of time to read it. vi Table of Contents List of Figures ix 1 Introduction1 2 Universal Algebra7 3 Basic Properties of 2-Semilattices 17 3.1 The Digraph................................... 19 3.2 Local Finiteness and Meet Semidistributivity................. 29 3.3 Absorption.................................... 33 3.4 Subdirect Products of Strongly Connected 2-Semilattices.......... 36 4 Congruence Lattices of Finite 2-Semilattices 43 4.1 Connectivity and Minimal Congruences.................... 44 4.2 Tame Congruence Theory in 2-Semilattices.................. 48 4.3 Maximal Congruences............................. 53 vii 5 Residually Large Varieties 61 5.1 Background................................... 61 5.2 Acyclic and Weakly Acyclic 2-Semilattices.................. 63 5.3 Acyclic and Weakly Acyclic Varieties are Residually Large......... 68 6 The Constraint Satisfaction Problem 77 6.1 Background................................... 77 6.2 The (2; 3)-Consistency Algorithm....................... 82 6.3 Bulatov's Algorithm for CSP over 2-Semilattices............... 88 6.4 Bulatov Solutions................................ 94 6.5 Maltsev Products Involving 2-Semilattices.................. 98 6.6 Extending Bulatov's Result.......................... 104 References 114 Index 120 viii List of Figures 1.1 The operation table for T3 ...........................3 3.1 The operation tables for A1 and A2 ...................... 21 3.2 The digraph of A1 and A2 ........................... 21 3.3 The digraph of E ................................ 27 3.4 A finitely-generated infinite 2-semilattice................... 30 4.1 Distinguishing the congruences in the proof of Theorem 4.2.3........ 53 4.2 The operation table of A from Example 4.3.5................ 56 4.3 The graph of B from Example 4.3.5...................... 57 4.4 The operation table of B from Example 4.3.5................ 58 4.5 The graph of B from Example 4.3.5...................... 60 6.1 The operation table for m in Example 6.6.6................. 111 ix Chapter 1 Introduction One of the broad goals in universal algebra is to understand and classify algebras and classes of algebras called varieties. One such coarse classification was given by Hobby and McKenzie in [29]. This six-fold classification of locally finite varieties (defined in Chapter2) has provided a popular way of restricting interesting problems to a more approachable context. One of these six classes is the class of varieties having the congruence meet semidistributive property, which will be defined in Chapter2. Perhaps the simplest congruence meet semidistributive variety is the variety of semilat- tices. A semilattice is a set A equipped with a binary operation ^ which is commutative, associative, and idempotent. By idempotent, we mean a ^ a = a for every a 2 A. An important feature of semilattices is their connection with partial orders. Indeed, every semilattice (A; ^) carries a natural partial order given by a ≤ b if and only if a ^ b = a. As well, every partially ordered set with greatest lower bounds gives rise to a semilattice. In this case, ^ is the binary operation on the domain of the partial order which selects the greatest lower bound. Semilattices have received attention from researchers in a variety of areas of mathematics. Anderson and Ward in [1] and Rhodes in [50] discuss semilattices 1 with topological motivation. Ellis [21], Papert [45], Nieminen [44], and Freese and Nation [26] have written about semilattices from are algebraic point of view. Jeavons et. al. [31] and Rehof and Torben [49] are papers on constraint satisfaction problems. Semilattices are well-studied and well-understood from many perspectives. For a survey of the theory of semilattices, see Chajda et. al [17]. From the perspectives of this thesis, semilattices are very easily understood. On the purely algebraic side, semilattices are simple in the sense that every semilattice embeds in a direct product of copies of the two-element semilattice. This was essentially shown for finite semilattices in [45] by Papert, but the proof given extends to the infinite case. When an algebra embeds in a product of finite algebras, it is called residually finite. Papert's result shows that semilattices are residually finite. Semilattices are also simple from the perspective of the constraint satisfaction or homomorphism problem. For a relational structure A with domain A, we say that A is invariant under a semilattice operation ^ on A if each relation of A is a subuniverse of some power of the algebra (A; ^). If A is a relational structure which is invariant under a semilattice operation, the problem of determining whether or not a relational structure B similar to A has a homomorphism to A is decidable in polynomial time. This was first shown using arc-consistency by Jeavons et. al. in [31]. For a more in-depth discussion of arc-consistency, see Chapter 11 of Dechter [20].
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