Lazy groupoids
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Lazy groupoids Balázs Kaprinai1 · Hajime Machida2 · Tamás Waldhauser1 Received: 2 May 2020 / Accepted: 8 June 2020 © The Author(s) 2020
Abstract A binary operation f(x, y) is said to be lazy if every operation that can be obtained from f by composition is equivalent to f(x, y), f(x, x) or x. We describe lazy opera‑ tions by identities (i.e., we determine all varieties of lazy groupoids), and we also characterize lazy groupoids up to isomorphism. Keywords Binary operation · Groupoid · Semigroup · Term operation · Clone · Variety · Lazy operation · Lazy groupoid
1 Introduction Given a (not necessarily associative) binary operation f (x, y) = xy , we can form many other operations by composing f by itself, such as (xy)z, ((xy)(zu))(u(yv)), x1 (x2 (x3 ⋯ (xn−1 xn ))) , etc. These composite operations can have arbitrarily many variables, but sometimes it happens that they do not depend on all of their variables. Consider, for example, a rectangular band, i.e., a semigroup satisfying the identi‑ ties xx ≈ x (idempotency) and xyz ≈ xz . These identities imply x1 x2 ⋯ xn ≈ x1 xn for all n ∈ ℕ , thus every product can be reduced to a product of at most two variables. It is natural to say that the multiplication of a rectangular band is lazy, since it only generates the operations f(x, y) and f(x, x) (up to renaming variables), and we can get these from f by simply identifying variables, hence composition is “unproductive” in this case. Communicated by Victoria Gould. * Tamás Waldhauser [email protected]‑szeged.hu Balázs Kaprinai [email protected] Hajime Machida [email protected] 1
Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged 6720, Hungary
2
Tokyo, Japan
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Motivated by this example, we shall say that a binary operation f on a set A is lazy, if the only operations that can be obtained from f by composition are f(x, y) and f(x, x). We will give a more precise definition of laziness for operations of arbitrary arities in the Sect. 2. The main goal of this paper is to describe all lazy binary opera‑ tions and the corresponding groupoids (A; f). In Sect. 3 we will characterize lazy groupoids by identities: we will prove that they fall into 15 varieties (Theorem 3.5). One of these varieties is the semigroup variety defined by (xy)z ≈ x(yz) ≈ xz , which contains rectangular bands as a subvariety. We will determine all subvarieties of the 15 maximal lazy groupoid varieties in Sect. 4 (Theorem 4.2). In Sect. 5 we give a more concrete description of lazy groupoids: we character‑ ize them up to isomorphism by explicitly constructing their multiplication tables. This description is similar in spirit to the well known construction of rectangular bands as groupoids of the form (A1 × A2 ;⋅) , where the multiplication is defined by (a1 , a2 ) ⋅ (b1 , b2 ) = (a1 , b2 ). Lazy operations were originally defined in [5] in connection with essentially minimal clones. The 15 varieties of lazy groupoids were described already in the conference paper [6] (but th
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