Advances in the Theory of Varieties of Semigroups

Description

This monograph thoroughly explores the development of the theory of varieties of semigroups and of two related algebras: involution semigroups and monoids. Through this in-depth analysis, readers will attain a deeper understanding of the differences between these three types of varieties, which may otherwise seem counterintuitive. New results with detailed proofs are also presented that answer previously unsolved fundamental problems. Featuring both a comprehensive overview as well as highlighting the author’s own significant contributions to the area, this book will help establish this subfield as a matter of timely interest. Advances in the Theory of Varieties of Semigroups will appeal to researchers in universal algebra and will be particularly valuable for specialists in semigroups. Preface Contents 1 Historical Overview and Main Results 1.1 Important Varietal Properties 1.1.1 Finite Basis Problem 1.1.2 Hereditary Finite Basis Property 1.1.3 Cross Varieties 1.2 Varieties Generated by Completely 0-Simple Semigroups 1.2.1 Rees–Suschkewitsch Varieties 1.2.2 The Varieties A2 and B2 1.2.3 Aperiodic Rees–Suschkewitsch Varieties 1.2.4 Rees–Suschkewitsch Varieties Containing Nontrivial Groups 1.2.5 Aperiodic Rees–Suschkewitsch Monoids 1.3 Hereditary Finite Basis Property 1.3.1 Hereditarily Finitely Based Identities 1.3.2 Pseudo-Simple Hereditarily Finitely Based Identities 1.3.3 Minimal Non-Finitely Based Semigroups 1.4 Non-Finite Basis Property 1.4.1 Finite Basis Problem for Finite Semigroups 1.4.2 Establishing the Non-Finite Basis Property 1.4.2.1 Critical Rees Matrix Semigroups 1.4.2.2 Inherently Non-Finitely Based Finite Semigroups 1.4.2.3 Syntactic Method 1.4.2.4 A Comparison of the Three Methods 1.4.3 Irredundant Identity Bases 1.5 Varieties of Involution Semigroups 1.5.1 Equational Properties of Involution Semigroups 1.5.2 Lattice of Varieties of Involution Semigroups 1.5.3 Relationship Between an Involution Semigroup and Its Semigroup Reduct 1.5.3.1 Non-Twisted Involution Semigroups 1.5.3.2 Inherent Non-Finite Basis Property 1.5.3.3 Sufficient Conditions for the Non-Finite Basis Property 1.6 Varieties of Monoids 1.6.1 Rees Quotients of Free Monoids 1.6.2 Limit Varieties and Hereditarily Finitely Based Varieties 1.6.3 Cross Varieties and Inherently Non-Finitely GeneratedVarieties 1.6.4 Further Examples Involving Rees Quotients of Free Monoids 2 Preliminaries 2.1 Identities and Deducibility 2.2 Varieties and Identity Bases 2.3 Connected Words and Identities 2.4 Rees Quotients of Free Monoids 2.5 Involution Semigroups 2.5.1 Terms, Words, and Plain Words 2.5.2 Identities and Deducibility Part I Semigroups 3 Aperiodic Rees–Suschkewitsch Varieties 3.1 Background Information on L(A2) 3.1.1 Identity Bases for Some Subvarieties of A2 3.1.2 Identities Defining Varieties in [A0,A2] 3.1.3 A Decomposition of L(A2-) 3.2 Finite Basis Property for Subvarieties of A2 3.2.1 Varieties in I2=[A0,B2-] 3.2.2 Varieties in I1=[A0vB2,A2-] 3.2.3 Varieties in I3=[B2,A0-] and I4=[B0,A0-B2-] 3.2.4 Varieties in I5=L(B0-) 3.3 The Lattice L(A2) 3.3.1 The Interval I5=L(B0-) 3.3.2 The Varieties Dl, E, F, i 3.3.2.1 The Varieties Dl 3.3.2.2 The Varieties E and F 3.3.2.3 The Varieties i 3.3.3 The Intervals I1=[A0vB2,A2-] and I2=[A0,B2-] 3.3.4 The Intervals I3=[B2,A0-] and I4=[B0,A0-B2-] 3.3.5 The Interval [B0,A2] 3.4 Subvarieties of A2 That Are Cross, Finitely Generated, or Small 3.4.1 Cross Subvarieties and Small Subvarieties of A2 3.4.2 Finitely Generated Subvarieties of A2 3.5 Summary 4 Pseudo-Simple Hereditarily Finitely Based Identities 4.1 Non-Homotypical Identities 4.2 Homotypical Identities 4.3 Summary 5 Sufficient Conditions for the Non-Finite Basis Property 5.1 Identities Satisfied by L3 5.2 Proof of Theorem 5.1 5.3 Specialized Versions of Theorem 5.1 5.4 Summary 6 Semigroups Without Irredundant Identity Bases 6.1 Sufficient Condition for the Nonexistence of Irredundant IdentityBases 6.2 Identities Satisfied by L3,n 6.3 Sandwich Identities 6.4 Restrictions on Sandwich Identities 6.4.1 Level of Sandwiches Forming Sandwich Identities 6.4.2 Refined Sandwich Identities 6.5 An Explicit Identity Basis for L3,n 6.6 Nonexistence of Irredundant Identity Bases for L3,n 6.7 Summary Part II Involution Semigroups 7 Involution Semigroups with Infinite Irredundant Identity Bases 7.1 Identities Satisfied by 7.2 Connected Identities and -Sandwich Identities 7.2.1 Connected Identities 7.2.2 *-Sandwich Identities 7.3 Restrictions on *-Sandwich Identities 7.3.1 Type of *-Sandwiches Forming *-Sandwich Identities 7.3.2 Refined *-Sandwich Identities 7.4 An Explicit Identity Basis for with R>1 7.4.1 An Identity Basis from (7.1) 7.4.2 A Simpler Identity Basis 7.5 An Infinite Irredundant Identity Basis for with R>1 7.5.1 The Identities (7.4j) 7.5.2 Proof of Theorem 7.21 7.6 Smaller Examples 7.7 Summary 8 Finitely Based Involution Semigroups with Non-Finitely Based Reducts 8.1 Identities and *-Sandwich Identities Satisfied by 8.2 Restrictions on *-Sandwich Identities 8.3 An Explicit Identity Basis for 8.4 A Finite Identity Basis for 8.5 Summary 9 Counterintuitive Examples of Involution Semigroups 9.1 Involution Semigroups with Different Types of Identity Bases 9.1.1 Involution Semigroups with an Irredundant Identity Basis 9.1.2 Involution Semigroups Without Irredundant Identity Bases 9.2 Two Incomparable Chains of Varieties of Involution Semigroups 9.2.1 The Involution Semigroups 9.2.2 The Involution Semigroups 9.2.3 Proof of Theorem 9.10 9.3 Summary 10 Equational Theories of Twisted Involution Semigroups 10.1 Organized Identity Bases 10.2 Proof of Theorem 10.1 10.3 Summary Part III Monoids 11 Hereditarily Finitely Based Varieties of Monoids 11.1 Identities Satisfied by Noncommutative Subvarieties of O 11.1.1 Canonical Form 11.1.2 Fundamental Identities and Well-Balanced Identities 11.1.3 Proof of Proposition 11.2 11.2 Finite Basis Property of Subvarieties of O 11.3 Distinguished Varieties 11.4 Summary 12 Varieties of Aperiodic Monoids with Central Idempotents 12.1 Rigid Identities 12.1.1 Definition and Basic Properties 12.1.2 Straubing Identities 12.1.3 Limiting Identities 12.2 The Variety K 12.2.1 Almost Cross Property 12.2.2 Subvarieties of K 12.3 Cross Subvarieties of Azen 12.4 Varieties Inherently Non-Finitely Generated Within Azen 12.5 A Non-Finitely Generated Subvariety of RQx2y2 12.6 Summary 13 Certain Cross Varieties of Aperiodic Monoids with Commuting Idempotents 13.1 The Variety Q1 13.2 Varieties that Contain Q1 13.3 Varieties that Exclude K 13.4 Proof of Theorem 13.1 13.5 Summary 14 Counterintuitive Examples of Monoids 14.1 The Direct Product of RQxyx with Noncommutative Groups of Finite Exponent 14.1.1 Identities Satisfied by RQxyx and by Noncommutative Groups 14.1.2 Proof of Theorem 14.1 14.2 Finitely Based Monoids from Non-Finitely Based Semigroups 14.3 Summary References List of Symbols List of Symbols General Symbols Aspects of a General Word w Words Identities Finite Algebras Varieties of Semigroups Varieties or Classes of Monoids Index