Subdirect products of free semigroups and monoids Ashley Clayton This thesis is submitted in partial fulfilment for the degree of Doctor of Philosophy (PhD) at the University of St Andrews April 2020 Abstract Subdirect products are special types of subalgebras of direct products. The purpose of this thesis is to initiate a study of combinatorial properties of subdirect products and fiber products of semigroups and monoids, motivated by the previous work on free groups, and some recent advances in general algebra. In Chapter 1, we outline the necessary preliminary definitions and results, including elements of algebraic semigroup theory, formal language theory, automata theory and universal algebra. In Chapter 2, we consider the number of subsemigroups and subdirect prod- ucts of N×N up to isomorphism. We obtain uncountably many such objects, and characterise the finite semigroups S for which N × S has uncountably many subsemigroups and subdirect products up to isomorphism. In Chapter 3, we consider particular finite generating sets for subdirect prod- ucts of free semigroups introduced as \sets of letter pairs". We classify and count these sets which generate subdirect and fiber products, and discuss their abundance. In Chapter 4, we consider finite generation and presentation for fiber products of free semigroups and monoids over finite fibers. We give a characterisation for finite generation of the fiber product of two free monoids over a finite fiber, and show that this implies finite presentation. We show that the fiber product of two free semigroups over a finite fiber is never finitely generated, and obtain necessary conditions on an infinite fiber for finite generation. i In Chapter 5, we consider the problem of finite generation for fiber products of free semigroups and monoids over a free fiber. We construct two-tape au- tomata which we use to determine the language of indecomposable elements of the fiber product, which algorithmically decides when they are finitely generated. Finally in Chapter 6, we summarise our findings, providing some further questions based on the results of the thesis. ii Declarations Candidate’s declaration I, Ashley Clayton, do hereby certify that this thesis, submitted for the degree of PhD, which is approximately 40 000 words in length, has been written by me, and that it is the record of work carried out by me, or principally by myself in collaboration with others as acknowledged, and that it has not been submitted in any previous application for any degree. I was admitted as a research student at the University of St Andrews in September 2016. Ireceivedfundingfromanorganisationorinstitutionandhaveacknowledged the funder(s) in the full text of my thesis. Date Signature of candidate Supervisor’s declaration IherebycertifythatthecandidatehasfulfilledtheconditionsoftheReso- lution and Regulations appropriate for the degree of PhD in the University of St Andrews and that the candidate is qualified to submit this thesis in application for that degree. Date Signature of supervisor iii Permission for publication In submitting this thesis to the University of St Andrews we understand that we are giving permission for it to be made available for use in accordance with the regulations of the University Library for the time being in force, subject to any copyright vested in the work not being a↵ected thereby. We also understand, unless exempt by an award of an embargo as requested below, that the title and the abstract will be published, and that a copy of the work may be made and supplied to any bona fide library or research worker, that this thesis will be electronically accessible for personal or research use and that the library has the right to migrate this thesis into new electronic forms as required to ensure continued access to the thesis. I, Ashley Clayton, confirm that my thesis does not contain any third-party material that requires copyright clearance. The following is an agreed request by candidate and supervisor regarding the publication of this thesis: Printed copy No embargo on print copy. Electronic copy No embargo on electronic copy. Date Signature of Candidate Date Signature of Supervisor iv Underpinning Research Data or Digital Outputs Candidate’s declaration I, Ashley Clayton, hereby certify that no requirements to deposit original research data or digital outputs apply to this thesis and that, where ap- propriate, secondary data used have been referenced in the full text of my thesis. Date Signature of candidate v Contents Abstract i 1 Introduction and preliminaries 1 1.1 Conventions and structure . .7 1.2 Semigroups, monoids and homomorphisms . .8 1.3 Relations, congruences, quotients of semigroups & monoids . 18 1.4 Green's relations on semigroups and monoids . 22 1.5 Formal language theory, automata, free semigroups & monoids 27 1.6 Generating and presenting semigroups and monoids . 30 1.7 Direct products, subdirect products and fiber products . 41 2 Subdirect products involving the free monogenic semigroup 50 2.1 Subsemigroups of N × N ..................... 52 2.2 Subsemigroups of direct products of N with a finite semigroup 64 2.3 Subdirect powers of N ...................... 70 2.4 Subdirect products of N with a finite semigroup . 72 3 Counting finitely generated subdirect products and fiber prod- ucts of free semigroups 78 3.1 Sets of letter pairs generating subdirect products of free semi- groups . 79 3.2 Sets of letter pairs generating fiber products of free semigroups 84 3.3 Proportion of sets of letter pairs generating subdirect products and fiber products . 91 vi 4 Finitary properties for fiber products of free semigroups and monoids 96 4.1 Finite generation for fiber products of free monoids over finite fibers . 98 4.2 Finite presentation for fiber products of free monoids over fi- nite fibers . 107 4.3 Finite generation for fiber products of free semigroups over infinite fibers . 118 5 Deciding finite generation for fiber products of free semi- groups and monoids 131 5.1 Equivalent conditions to finite generation for fiber products of free semigroups and monoids over free fibers . 133 5.2 Two-tape automata construction for fiber products of free monoids over free monoid fibers . 136 5.3 The language recognised by the two-tape automaton associ- ated with Π('; )........................ 144 5.4 Decidability of finite generation for fiber products of free semi- groups and monoids over free fibers . 158 6 Concluding remarks and further questions 163 Bibliography 167 Acknowledgements 170 vii Chapter 1 Introduction and preliminaries In its operational form, the algebraic theory of semigroups is in part the developing abstraction of the algebraic theory of groups. The latter theory has of course been well studied and researched as early as the late 18th cen- tury. Whereas elements of group theory have formalised the notions of sym- metries for mathematical objects, semigroup theory attempts to formalise notions of partial symmetries and transformations of those objects. A par- ticularly notable example of this is the Wagner-Preston theorem ([16, The- orem 5.1.7]), which states that any inverse semigroup (called \generalised" groups, by Wagner [29]) is embeddable in a symmetric inverse semigroup (a \generalised" symmetric group). The symmetric inverse semigroup is a structure of partial symmetries (see [16, Theorem 5.1.5]), and so the Wagner- Preston theorem gives the partial symmetry analogue to Cayley's theorem for groups. Though the origins of the theory of semigroups can be traced back to the early twentieth century in the papers of Dickson [11] and de S´eguier[10], it has been argued that its research beginnings can be attributed with Sushkevich [28] in the late 1920s, and the celebrated paper of Rees [25] in 1940. This makes it a relatively recent and modern study in the history of Algebra, and Mathematics in general. In this author's opinion, the theory benefits from its youth with a metaphorical continent of roads untravelled. Though it 1 comprises many interesting open problems in its own right, it further provides many theoretical techniques and insights into other algebraic studies, such as group theory. Moreover, it has seen practical application via the related study of theoretical computer science, which is another relatively recent field. One closely linked brethren theory to that of semigroup theory is the study of finite automata and formal languages. This study encompasses the theoretical formalisation of algorithmic decision making machines, such as finite state automata, transducers, and Turing machines. There is a plethora of practical computing application in this field, though its formulisation lends itself to the study of pure semigroup theory particularly well. Strings of characters can be viewed as abstract semigroup products of elements referred to as letters, and vice versa. These products can be viewed as words over the alphabet of the possible characters for a string. The set of all such words form languages, which can be recognised by the machinery of automata, which can decide if a given word is in a language or not. As such, problems of deciding properties of semigroup elements have been formulated as decision problems in the theory of automata. Though this work comprises many elements of the pure and applied studies outlined above, it is primarily motivated from the perspective of pure semi- group theory. Its nature then will be to investigate, develop, and advance theoretical concepts of semigroup theory from the metaphorically aforemen- tioned roads less travelled. In this work, that road will be the theory of subdirect products of semigroups. Heuristically, in the more general theory of universal algebra, subdirect products are special types of subalgebras of direct products of algebras. Each direct factor can in some way be embedded into the subdirect product, making the factors of the subdirect product\full" in some sense. As substructures of direct products, subdirect products in a way generalise the algebraic construction of direct products.