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On Homology of Associative Shelves Mathematics Faculty Works Mathematics 2017 On Homology of Associative Shelves Alissa Crans Loyola Marymount University, [email protected] Follow this and additional works at: https://digitalcommons.lmu.edu/math_fac Part of the Mathematics Commons Recommended Citation Crans, Alissa, et al. “On Homology of Associative Shelves.” Journal of Homotopy & Related Structures, vol. 12, no. 3, Sept. 2017, pp. 741–763. doi:10.1007/s40062-016-0164-9. This Article is brought to you for free and open access by the Mathematics at Digital Commons @ Loyola Marymount University and Loyola Law School. It has been accepted for inclusion in Mathematics Faculty Works by an authorized administrator of Digital Commons@Loyola Marymount University and Loyola Law School. For more information, please contact [email protected]. J. Homotopy Relat. Struct. (2017) 12:741–763 DOI 10.1007/s40062-016-0164-9 On homology of associative shelves Alissa S. Crans1 · Sujoy Mukherjee2 · Józef H. Przytycki2,3 Received: 1 April 2016 / Accepted: 15 November 2016 / Published online: 19 December 2016 © Tbilisi Centre for Mathematical Sciences 2016 Abstract Homology theories for associative algebraic structures are well established and have been studied for a long time. More recently, homology theories for self- distributive algebraic structures motivated by knot theory, such as quandles and their relatives, have been developed and investigated. In this paper, we study associative self-distributive algebraic structures and their one-term and two-term (rack) homology groups. Keywords Spindles · Knot theory · One-term and two-term (rack) distributive homology · Unital · Self-distributive semigroups · Laver tables Communicated by Vladimir Vershinin. Alissa S. Crans was supported by a grant from the Simons Foundation (#360097, Alissa Crans). Sujoy Mukherjee was supported by the Presidential Merit Fellowship of the George Washington University. Józef H. Przytycki was partially supported by the Simons Foundation Collaboration Grant for Mathematicians-316446. B Józef H. Przytycki [email protected] Alissa S. Crans [email protected] Sujoy Mukherjee [email protected] 1 Department of Mathematics, Loyola Marymount University, Los Angeles, CA, USA 2 Department of Mathematics, The George Washington University, Washington DC, USA 3 University of Gda´nsk, Gda´nsk,Poland 123 742 A. S. Crans et al. Mathematics Subject Classification Primary 18G60; Secondary 20M32, 20N02, 57M25 Self-distributive structures1 have been studied for over a century. In 1880 Peirce [20] emphasized the importance of right self-distributivity in algebraic structures, when his friend Schröder [27] constructed an example of a non-associative three element magma. Two-sided self-distributive structures were studied in 1929 by Burstin and Mayer [2]; compare [23]. In the past several decades, self-distributive structures have secured an important role in knot theory. In May of 2015, during a discussion of the origin of the word ‘quandle’ on the n-Category Café (https://golem.ph.utexas.edu/category/), Sam C.2 remarked that uni- tal shelves are associative and provided a short elegant proof. The work herein was inspired by this post. In Sect. 1, we provide definitions and discuss some elementary propositions. The first two of these propositions were proven by Sam C. [26]. The latter propositions lay the pathway for the subsequent sections. Section 2 addresses finitely generated associative shelves, their cardinalities and generating functions. Following this, finitely generated proto-unital shelves, pre-unital shelves, and unital shelves are discussed. While some of the observations in this section are new, others are stated for completeness [11,12,29]. In Sect. 3, the two well-known homology theories for self-distributive algebraic structures, namely one-term distributive homology and two-term (rack) homology are defined and the main results concerning the homology groups of a special class of associative shelves are proven. In particular, we compute one-term and two-term (rack) homology of unital shelves. The main theorem concerning two-term homology of shelves with right fixed elements extends to the two-term (rack) homology groups of Laver tables [4,5,16] and f-block spindles [3,24]. Henceforth, we will use the term two-term homology to refer to rack homology. Finally, Sect. 4 and the Appendix contain conjectures and computational data. 1 Introduction Definition 1.1 A quandle (X, ∗) is a magma3 satisfying the following properties: (1) (idempotence) a ∗ a = a for all a ∈ X, (2) (inverse) There exists ∗:¯ X × X −→ X such that (a ∗¯ b) ∗ b = a = (a ∗ b) ∗¯ b for all a, b ∈ X, and (3) (self-distributivity) (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c) for all a, b, c ∈ X The three properties satisfied by the elements of a quandle are the algebraic ana- logues of the conditions satisfied by quandle colorings under the three Reidemeister moves, depicted in Fig. 1. The convention for coloring arcs of a knot diagram by elements of a quandle X is illustrated in Figs. 2 and 3. The primordial example of a 1 The word distributivity was coined in 1814 by French mathematician François Servois (1767–1847). 2 Unfortunately we do not know his full name. In [26], he writes: “I am not a mathematician or physicist...”. 3 The term ‘magma’ was introduced by Jean-Pierre Serre in 1965. The older term ‘groupoid,’ introduced by Øystein Ore in 1937, now has different meaning: it is a category in which every morphism is invertible. 123 On homology of associative shelves 743 Fig. 1 Unoriented Reidemeister moves; coherent orientations could be added R2 R1 R3 Fig. 2 Quandle coloring rules a b b a _ b a*b a*b b Fig. 3 Reidemeister move 3 abc a b c and quandle coloring b*c a*b a*c c b*c (a*c)*(b*c) c b*c (a*b)*c quandle arises when starting with a group (G, ·) and defining the quandle operation as a ∗ b = b−1 · a · b for all a, b ∈ G. Definition 1.2 By relaxing some of the axioms of a quandle, we obtain generalizations of quandles, including: (1) A rack is a magma which satisfies the second and third quandle axioms. (2) A spindle is a magma which satisfies the first and the third axioms of a quandle. (3) A shelf is a magma which satisfies only the third quandle axiom. (4) A unital shelf is a shelf (X, ∗) equipped with an element 1, such that x ∗ 1 = x = 1 ∗ x for all x ∈ X. 123 744 A. S. Crans et al. Table 1 An associative spindle ∗ abcd a aacd b abcd c aacd d adcd Table 1 depicts an associative spindle that is not a quandle. We continue with some basic observations, due to Sam C. Proposition 1.3 (Sam C.) Let (X, ∗) be a unital shelf. Then: (0) a ∗ a = a for all a ∈ X (1) a ∗ b = b ∗ (a ∗ b) for all a, b ∈ X (2) a ∗ b = (a ∗ b) ∗ b for all a, b ∈ X Proof Let a, b ∈ X, where X is a unital shelf. Then: (0) a = (1 ∗ 1) ∗ a = (1 ∗ a) ∗ (1 ∗ a) = a ∗ a (1) a ∗ b = (1 ∗ a) ∗ b = (1 ∗ b) ∗ (a ∗ b) = b ∗ (a ∗ b) (2) a ∗ b = (a ∗ 1) ∗ b = (a ∗ b) ∗ (1 ∗ b) = (a ∗ b) ∗ b Remark 1.4 (1) Monoids satisfying condition (1) of Proposition 1.3 are known as ‘graphic monoids’ and have been studied in [13,28].4 (2) Semi-groups satisfying b∗(a∗b) = a∗b are called graphic semi-groups.Itturns out graphic semi-groups are shelves because (a ∗ c) ∗ (b ∗ c) = a ∗ (c ∗ (b ∗ c)) = a ∗ (b ∗ c) = (a ∗ b) ∗ c. In particular, graphic monoids are self-distributive so that unital shelves and graphic monoids are the same algebraic structures. (3) In Proposition 1.3, the proofs of parts (0) and (1) require only the left-sided unity, whereas for the proof of part (2), both the left-sided and right-sided units are needed. Definition 1.5 A shelf satisfying axioms (0), (1), and (2) of Proposition 1.3 is called a pre-unital shelf and a shelf satisfying properties (1) and (2) of Proposition 1.3 is called a proto-unital shelf. Corollary 1.6 (Sam C.) Proto-unital shelves are associative. In particular, this implies unital shelves are associative. Proof Let a, b, c ∈ X, where X is a proto-unital shelf. Then: (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c) = (a ∗ (b ∗ c)) ∗ (c ∗ (b ∗ c)) = (a ∗ (b ∗ c)) ∗ (b ∗ c) = a ∗ (b ∗ c) 4 In [13], the number of elements in the free graphic monoid was computed. The term graphic monoid was coined by Lawvere [15]. 123 On homology of associative shelves 745 Table 2 A non-associative quasi-group quandle ∗ abcd a acdb b dbac c bdca d cabd We conclude by illustrating a relationship between pre-unital and unital shelves. Proposition 1.7 If (X, ∗) is a pre-unital shelf, then (X ∪{1}, ∗) is unital. Conversely, if (X, ∗) is a unital shelf, then (X\{1}, ∗) is pre-unital. Proof Extending a pre-unital shelf to a unital shelf requires verifying the self- distributivity property for all a, b, c ∈ X where at least one of these elements is the unit. This follows easily. Conversely, for two elements a, b ∈ X, neither of which is the unit, a ∗ b = 1. Namely, if a ∗ b = 1 then (a ∗ b) ∗ b = b. Thus by condition (2) of Proposition 1.3, a ∗ b = b, and b = 1. Therefore, the result follows. In the sections that follow, we study various properties of associative shelves, but not of associative racks and quandles, even though these two algebraic structures are more important from a knot-theoretic point of view. This is due to the fact that associative racks are trivial. If (X, ∗) is an associative rack with a, b ∈ X, then (a ∗ b) ∗ b = (a ∗ b) ∗ (b ∗ b).
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