Lazy groupoids

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 operations by identities (i.e., we determine all varieties of lazy groupoids), and we also characterize lazy groupoids up to isomorphism.


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)), x 1 (x 2 (x 3 ⋯ (x n−1 x n ))) , 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 identities xx ≈ x (idempotency) and xyz ≈ xz . These identities imply x 1 x 2 ⋯ x n ≈ x 1 x n 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.

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Lazy groupoids 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 operations 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 characterize 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 (A 1 × A 2 ;⋅) , where the multiplication is defined by (a 1 , a 2 ) ⋅ (b 1 , b 2 ) = (a 1 , b 2 ).
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 the proof of Theorem 3.5 was only sketched there), and then an application to essentially minimal clones was given. Thus, the present paper can be regarded as an extended version of [6], the new contributions being the determination of all (sub)varieties of lazy groupoids and the explicit description of lazy groupoids up to isomorphism.

Preliminaries
An n-ary operation on a nonempty set A is a map f ∶ A n → A . We denote the set of n-ary operations on A by O (n) A , and O A stands for the set of all finitary operations on A. We say that the i-th variable of f ∈ O (n) A is essential (in other words, f depends on its i-th variable) if there exist a 1 , … , a i , a � i , … , a n ∈ A such that For 1 ≤ i ≤ n ∈ ℕ we define the i-th n-ary projection e (n) A clone is a set C ⊆ O A of operations that is closed under composition and contains every projection. The clone generated by a given operation f is the clone [f ] containing all operations that can be obtained from f and the projections by composition. Equivalently, [f ] is the clone of term functions of the algebra = (A;f ) . If f is a binary operation, then we will use the notation f (x, y) = x ⋅ y = xy , and then the algebra = (A;f ) = (A;⋅) is called a groupoid. For the sake of simplicity, let us say that the groupoid is essentially binary (essentially at most unary) if f depends on both of its variables (f depends on at most one variable).

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This means that g can be obtained from f by identifying variables, permuting variables and introducing inessential variables. The relation ⪯ gives rise to a quasiorder on O A . The corresponding equivalence relation is defined by f ≡ g ⟺ f ⪯ g and g ⪯ f , and it is clear that f ≡ g if and only if they differ only in inessential variables and/or in the order of their variables. Note that for any f ∈ O A , we have f ≡ id if and only if f ∈ J A . We use the notation ↓ f for the principal ideal (downset) generated by f in the subfunction quasiorder: Note that the set ↓ f contains only one unary operation, namely f (x, … , x) . For more information on the minor quasiorder and its principal ideals, see [4].
we say that f is a lazy operation and [f ] is a lazy clone. Thus f is a lazy operation if it does not generate any other operations but its identification minors and projections.
Indeed, if f is lazy, then f 2 ∈ J A ∪ ↓ f , and the latter set contains only two operations up to equivalence, namely id and f. Thus we have ). Conversely, each one of the given identities implies that [f ] = {id, f } , and thus f is lazy.
Lazy idempotent operations can be constructed as follows. Let A 1 , … , A n be nonempty sets, and let us define an n-ary operation f on A 1 × ⋯ × A n by for all a j i ∈ A i (i, j = 1, 2, … , n) . Note that the algebra (A 1 × ⋯ × A n ;f ) is the direct product of the algebras A i ;e (n) i (i = 1, 2, … , n) . These algebras were called n-dimensional diagonal algebras in [7] and n-ary rectangular bands in [6] (cf. the construction of binary rectangular bands in Sect. 1). It was shown in [6,7] that an idempotent operation f ∈ O (n) A is lazy if and only if (A; f) is (isomorphic to) an n-ary rectangular band.

Remark 2.3 For f ∈ O (n)
A and k ∈ {1, 2, … , n} , let f k ∈ O (2n−1) A denote the function obtained from f by substituting f for its k-th variable; more precisely, A is lazy then f k ⪯ f or f k ∈ J A for every k ∈ {1, 2, … , n} . This is a simple necessary condition for laziness, and it will serve as a starting point for our investigation of lazy binary operations. Note, however, that this condition is not sufficient for laziness, as it is shown by the following example. Let us consider the binary operation f (x, y) = xy on the set A = {0, 1, 2} that is defined by the multiplication table below. (1) , … , x (n) ).

Lazy groupoids
One can verify that this groupoid satisfies the identities (xy)z ≈ z 2 and x(yz) ≈ x 2 , which means that the functions f 1 (x, y) = (xy)z and f 2 (x, y) = x(yz) are both minors of f. However, f is not a lazy operation, because g(x, y, z, u) ∶= (xy)(zu) is constant 1, and g ∉ J A ∪ ↓ f .

Characterizing lazy groupoids by identities
Let f (x, y) = x ⋅ y = xy denote a binary operation on an arbitrary nonempty set A, and let denote the groupoid (A;f ) = (A;⋅) . The dual of is the groupoid d = (A;g) , where g(x, y) = yx , and the dual of a groupoid variety V is V d = d ∶ ∈ V . Clearly, a groupoid is lazy if and only if its dual is lazy.
If f depends only on at most one variable, then we have either f (x, y) = g(x) or f (x, y) = g(y) for some unary operation g. According to Example 2.1, f is lazy if and only if g satisfies either g(g(x)) ≈ g(x) or g(g(x)) ≈ x . This yields the following description of essentially at most unary lazy groupoids.

Lemma 3.1 If is a lazy essentially at most unary groupoid, then belongs to one of the following four varieties:
Proof The fact that f does not depend on its second variable can be expressed by the identity xy ≈ xz . Using the unary operation g as above, g(g(x)) ≈ g(x) translates to (xy)z ≈ xy and g(g(x)) ≈ x translates to (xy)z ≈ x , yielding the varieties U and Ũ . If f does not depend on its first variable, then we obtain the dual varieties U d and Ũ d . ◻ In the sequel, we will assume that f depends on both of its variables. We have f 1 (x, y) = (xy)z and f 2 (x, y) = x(yz) ; see Remark 2.3. If f is a lazy operation then f 1 , f 2 ∈ J A ∪ ↓ f , hence satisfies the identities (xy)z ≈ t 1 and x(yz) ≈ t 2 for some choice of the terms t 1 , t 2 ∈ x, y, z, x 2 , y 2 , z 2 , xy, yx, yz, zy, xz, zx . This gives us 144 possibilities; we will prove that only 20 of these are possible. Examining these cases, we will find that essentially binary lazy groupoids belong to 13 varieties, each being defined by two identities.
Proof The identity (xy)z ≈ t 1 (x, y, z) implies If t 1 = x , then we obtain xy ≈ xu , which shows that xy does not depend on y. If t 1 = zy , then we get uz ≈ (zy)u , which yields uz ≈ uy after applying (zy)u ≈ t 1 (z, y, u) , hence f does not depend on its second variable.
Let us now consider the case t 1 = yx . Then we have which immediately implies (vw)(xy) ≈ xy. On the other hand, we have (vw)(xy) ≈ t 1 (v, w, xy) ≈ wv . Thus xy ≈ wv , i.e., f is a constant operation. Similar arguments work for the remaining three cases; we summarize them in Table 1 of "Appendix 1". The identities x(yz) ≈ t 2 are the duals of the above ones. ◻ Now we are left with 36 pairs (t 1 , t 2 ) ; these possibilities are summarized in Table 2. We will prove in the next two lemmas that the entries marked by '−' contradict the assumption that f is essentially binary, while the other cases give rise to 7 varieties L 1 , … , L 7 of lazy groupoids together with their duals (note that L 7 is selfdual).

Lemma 3.3 Let be an essentially binary groupoid. If is lazy, then it belongs to one of the 13 varieties
which are defined by the following identities: Proof We can derive the following three identities from (xy)z ≈ t 1 and x(yz) ≈ t 2 : In all the 16 cases marked by '−' in Table 2, at least one of the above three identities contradicts the essentiality of the operation f. We work out the details only for the case t 1 ≈ y 2 , t 2 ≈ xy (here we will need the identity (3.1c)); the other cases are similar or even simpler (see Table 3): Now it only remains to verify the entries marked by L 1 ∩ L d 1 in Table 2. These can be handled with the help of the identities (3.1); again, we provide details only for one case, namely for t 1 ≈ y 2 , t 2 ≈ x 2 , and refer to Table 3 for the remaining cases. Note that the variety L 1 ∩ L d 1 is axiomatized by the identities (xy)z ≈ x(yz) ≈ x 2 ≈ z 2 . This means that a groupoid belongs to L 1 ∩ L d 1 if and only if is a semigroup and there is a constant c ∈ A such that xyz ≈ x 2 ≈ c . It is clear that such semigroups satisfy (xy)z ≈ y 2 and x(yz) ≈ x 2 . Conversely, assume now that (xy)z ≈ y 2 and x(yz) ≈ x 2 hold in a groupoid . Let us write out (3.1b): We can conclude that (yz) 2 depends neither on y nor on z, hence there is a constant c ∈ A such that (yz) 2 ≈ c . Now let us use (3.1a): Table 2 is special in the sense that the identities (xy)z ≈ z 2 , x(yz) ≈ x 2 do not guarantee laziness (see the example in Remark 2.3). Here (3.1a) yields (zu) 2 ≈ (xy) 2 , i.e., (xy) 2 is constant. Since the only constant in In order to complete the description of lazy groupoids, we still need to verify that every groupoid in the varieties L 1 , … , L 7 , L d 1 , … , L d 6 defined in Lemma 3.3 is indeed lazy. In the following, whenever we use one of the two defining identities for any one of our varieties, we write " 1.
≈ " to indicate whether we have used the first or the second identity (as listed in Lemma 3.3).

Lemma 3.4 If a groupoid belongs to one of the
Proof Assume that is a groupoid in L 4 (the proof for the other varieties is very similar; see Table 4). We prove by term induction that every term of is equivalent to x or xy (allowing that x and y are the same variable). Let t be a term that contains at least two multiplications (i.e., at least three, not necessarily distinct variables). Then t = s 1 ⋅ s 2 , where the terms s 1 and s 2 are shorter than t, hence, by the induction hypothesis, they are equivalent to a variable or to a product of two variables. Therefore, we have the following three possibilities with some (not necessarily distinct) variables x, y, z, u:

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Thus, every term over L 4 is indeed equivalent to a variable or a product of two variables, showing that every member of L 4 is a lazy groupoid. ◻

Theorem 3.5 A groupoid is lazy if and only if it belongs to one of the 15 varieties
Proof For essentially binary groupoids the "only if" part is covered by Lemma 3.3, while the "if" part follows from Lemma 3.4 and its dual. For essentially unary groupoids we can use Lemma 3.1. We do not need to list U and U d , since U ⊆ L i for every i (cf. Fig. 1). ◻

Varieties of lazy groupoids
It is easy to verify that the proper subvarieties of U, Ũ and their duals are the varieties LZ (left zero semigroups), RZ (right zero semigroups), Z (zero semigroups) and T (trivial semigroups), as shown in Fig. 1: Having determined all varieties of essentially at most unary lazy groupoids, let us now deal with subvarieties of L 1 , … , L 7 . We get such subvarieties by adding some extra identities to the defining two identities of L i (i = 1, … , 7) . Let V be the intersection of L i and the variety defined by the identity p ≈ q (we assume p ≠ q ). Laziness of L i implies that p and q are both equivalent to a product of two (not necessarily distinct) variables over V. If one of p and q involves two different variables, say p = xy , but q does not involve both x and y, then p ≈ q implies that f is essentially at most unary, hence V is a subvariety of U, U d , Ũ or Ũ d . If x and y both occur in q, then we get xy ≈ yx (commutativity). The remaining cases, when at most one variable occurs on both sides, are the following: x 2 ≈ y 2 (the main diagonal of the multiplication table is constant), x 2 ≈ y (satisfied only by trivial groupoids), x 2 ≈ x (idempotency), x ≈ y (satisfied only by trivial groupoids). Thus, in order to determine all non-unary subvarieties of L i , we need to compute the intersection of L i by one or more of the following three varieties: ≈ xz;

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Lazy groupoids exception of the 3 varieties L 1 ∩ D , L 1 ∩ C and L 7 ∩ I = RB (here, and in the sequel, RB denotes the variety of rectangular bands).
Proof We consider only L 3 ; the other cases can be seen in Table 5. Recall that L 3 is defined by (xy)z ≈ xy and x(yz) ≈ x 2 . As before, we will use 1.
≈ when we use one of these two identities, and we write ≈ (xy) 2 D ≈ x 2 , which means again that xy does not depend on y. Finally, in L 3 ∩ C we can deduce Proof By Theorem 3.5 every lazy groupoid belongs to L 1 , … , L 7 , U,Ũ or to the duals of these varieties. We have already determined the subvarieties of U and Ũ , and Lemma 4.1 describes all subvarieties of L 2 , … , L 7 (and their duals). Lemma 4.1 also implies that for L 1 and its dual we need to consider Therefore, the only variety that could be possibly missing from Fig. 1 is L 1 ∩ D ∩ C . However, this coincides with L 1 ∩ C , as L 1 ∩ C ⊆ L 1 ∩ D . Indeed, we can derive x 2 ≈ y 2 from commutativity and the two defining identities of L 1 : To prove that the 24 varieties in Fig. 1 are all distinct, we give the operation tables of their two-generated free algebras in "Appendix 2" (we include only one member of each pair of dual varieties, and we omit T). It remains to prove the containments indicated in Fig. 1. All of these are straightforward to verify, with the exception of L 1 ∩ C ⊆ L 1 ∩ D , which we have already proved. ◻

Characterizing lazy groupoids up to isomorphism
We will give a concrete description of groupoids in the varieties L 1 , … , L 7 in this section (we do not write out the details for the dual varieties L d 1 , … , L d 6 , and we also ignore the trivial unary cases Ũ and Ũ d ).
Let us start with the variety L 7 , which is the semigroup variety defined by xyz ≈ xz . If ∈ L 7 , then the set E of the idempotent elements in forms a rectangular band. We will prove below in Theorem 5.1 that is an inflation of the rectangular band = (E;⋅) . This means that each idempotent e ∈ E has a ≈ y 2 .
"neighborhood" N e containing e such that N e ∶ e ∈ E is a partition of A (in particular, the only idempotent in N e is e), and for each a ∈ N e , b ∈ N f we have ab = ef . In other words, can be constructed from the rectangular band by "inflating" each element e ∈ E to a set N e . Together with the well-known characterization of rectangular bands as direct products of a left zero semigroup and a right zero semigroup, this provides an explicit description of the members of L 7 . This result appeared in [1] (and perhaps it has been known even earlier), but we include the proof for completeness.

Theorem 5.1 A groupoid belongs to the variety L 7 if and only if there is a subset E ⊆ A and a partition N e ∶ e ∈ E , such that = (E;⋅) is a rectangular band, and
Proof Suppose that ∈ L 7 , let E = e ∈ A ∶ e 2 = e and let N e = a ∈ A ∶ a 2 = e .
Clearly, the sets N e (e ∈ E) are pairwise disjoint and nonempty (since e ∈ N e ); moreover, for all a ∈ A we have (a 2 ) 2 = (aa)(aa) 1.
= a 2 , hence e ∶= a 2 is idempotent and a ∈ N e . This shows that N e ∶ e ∈ E is indeed a partition of A. The set E is closed under multiplication, as (ef ) 2 = (ef )(ef ) = ef for all e, f ∈ E . Therefore (E;⋅) is a subsemigroup, and it satisfies the identities xyz ≈ xz and x 2 ≈ x , hence it is a rectangular band. It only remains to prove ab = ef for all a ∈ N e , b ∈ N f : Now assume that E ⊆ A such that = (E;⋅) is a rectangular band, and N e ∶ e ∈ E is a partition of A such that (5.1) holds. In order to verify that satisfies the identity (xy)z ≈ xz , let us consider arbitrary elements a, b, c ∈ A . Then a ∈ N e , b ∈ N f and c ∈ N g for some e, f , g ∈ E . From the second equality of (5.1) it follows that ab = ef and ac = eg . Since E is closed under multiplication, h ∶= ef belongs to E, and h 2 = h , as is a band. The first equality of (5.1) implies that h ∈ N h , and then we have hc = hg by the second equality of (5.1). Putting everything together, we obtain (ab)c = ac: where in the equality marked by RB we used the assumption that is a rectangular band. We have proved that satisfies the identity (xy)z ≈ xz , and the identity x(yz) ≈ xz can be verified in a similar way, proving that ∈ L 7 . ◻ For the varieties L 1 , … , L 6 , we will give similar characterizations in the following six theorems. Groupoids belonging to L 6 were described in [3] as unions of constant semigroups, which is essentially the same as Theorem 5.9 below, but our proof is different. Semigroups satisfying xyz ≈ xy were investigated and characterized in [2]; our Theorem 5.8 gives a different (and perhaps simpler) description.

Lazy groupoids
The characterizations will be given in terms of a partition N e ∶ e ∈ E similarly to Theorem 5.1, but we will also need to specify a subset S e ⊆ N e for all e ∈ E . Let us fix the notation for later reference: Notation 5.2 Let E ⊆ A , and let N e ∶ e ∈ E be a partition of A; moreover, for every e ∈ E , let S e ⊆ N e such that e ∈ S e .

Remark 5.3
Most of the time (with the exception of L 5 ), the setup will be the same as in Theorem 5.1: E will be the set of idempotents, N e = a ∈ A ∶ a 2 = e and S e will be the set of factorizable elements in N e , i.e., S e = N e ∩ {ab ∶ a, b ∈ A} . To see that N e ∶ e ∈ E is indeed a partition of A, it suffices to note that (xx)(xx) ≈ xx holds in L 1 , … , L 6 (by the same argument as in the second sentence of the proof of Theorem 5.1). This implies that the square of any element is idempotent, therefore the sets N e cover A.

Theorem 5.4 A groupoid belongs to the variety L 1 if and only if there is a parti-
tion N e ∶ e ∈ E as in Notation 5.2, such that (5.2) for all e, f ∈ E and a ∈ N e , b ∈ N f , we have a 2 = e and Fig. 1 The semilattice of lazy groupoid varieties Proof First assume that ∈ L 1 and recall that L 1 is defined by (xy)z ≈ x 2 and x(yz) ≈ x 2 . Let us consider the partition described in Remark 5.3; then the first equality of (5.2) is automatically satisfied. For the other four statements, let a ∈ N e and b ∈ N f . We have (ab) 2 = (ab)(ab)

1.
= a 2 = e , thus ab ∈ N e . Therefore ab ∈ S e , as ab is obviously factorizable, and this verifies the last statement of (5.2). If b ∈ S f , i. e., b = b 1 b 2 for some b 1 , b 2 ∈ A , then ab = a(b 1 b 2 ) 2.
Conversely, suppose that N e ∶ e ∈ E is a partition of A as in Notation 5.2, such that (5.2) holds. Let us compute (ab)c and a(bc) for a ∈ N e , b ∈ N f , c ∈ N g . From (5.2) we see that ab ∈ S e and bc ∈ S f , and then, again from (5.2), we obtain This shows that satisfies (xy)z ≈ x(yz) ≈ x 2 , hence ∈ L 1 . ◻ Theorem 5.4 allows us to construct the multiplication table of any groupoid in L 1 as follows. Fix an arbitrary nonempty set A, and choose a partition N e ∶ e ∈ E of A as in Notation 5.2. Then define a multiplication on A such that (5.2) is satisfied (if a ∈ N e ⧵ S e and b ∈ N f ⧵ S f , then we can choose ab to be any element of S e ). This gives a groupoid in L 1 , and every member of L 1 can be obtained this way. A part of such an operation table can be seen in "Appendix 3". Only two blocks of the partition are displayed; the elements of S e are denoted by e = s 0 , s 1 , … and the elements of N e ⧵ S e are denoted by a 1 , a 2 , … (and similarly for S f and N f ⧵ S f ). However, this is only for notational convenience: these sets can have arbitrary cardinalities (not necessarily countable). "Appendix 3" contains similar tables for the varieties L 2 , … , L 6 , illustrating the following five theorems.

Theorem 5.5 A groupoid belongs to the variety L 2 if and only if there is a partition N e ∶ e ∈ E as in Notation 5.2, such that
Proof First assume that ∈ L 2 and recall that L 2 is defined by (xy)z ≈ x 2 and x(yz) ≈ xy . Let us consider the partition described in Remark 5.3. Let a ∈ N e and b ∈ N f . We have (ab) 2 = (ab)(ab)

1.
= a 2 = e , hence ab ∈ S e . Moreover, ab   for all e, f ∈ E and a ∈ N e , b ∈ N f , we have a 2 = e and

Lazy groupoids
Conversely, suppose that N e ∶ e ∈ E is a partition of A as in Notation 5.2, such that (5.3) holds. Let us compute (ab)c and a(bc) for a ∈ N e , b ∈ N f , c ∈ N g . From (5.3) we see that ab ∈ S e and bc ∈ S f , and then, again from (5.3), we obtain This shows that satisfies (xy)z ≈ x 2 and x(yz) ≈ xy , hence ∈ L 2 . ◻ Theorem 5.6 A groupoid belongs to the variety L 3 if and only if there is a partition N e ∶ e ∈ E as in Notation 5.2, such that Proof First assume that ∈ L 3 and recall that L 3 is defined by (xy)z ≈ xy and x(yz) ≈ x 2 . Let us consider the partition described in Remark 5.3. Let a ∈ N e and b ∈ N f ; then (ab)(ab)

1.
= ab , which means that every product is idempotent (this already proves the last statement of (5.4)). In particular, if a = a 1 a 2 ∈ S e , then a 2 = a . However, the construction of the partition implies that a 2 = e , thus we can conclude that a = e . (Note that this means that S e = {e} .) Now we can write ab as ab = (a 1 a 2  This proves that satisfies (xy)z ≈ xy and x(yz) ≈ x 2 , hence ∈ L 3 . ◻ In the first half of the proof we observed that if the partition is chosen as in Remark 5.3, then each S e is a singleton. However, if we choose an arbitrary partition as in Notation 5.2 and we define the multiplication according to (5.4), then the resulting groupoid will be in L 3 . This is not a contradiction: given such an operation table (like the one in "Appendix 3"), we can redefine the set S e so that S e = {e} (i.e., we "move" all elements of S e to N e ⧵ S e except for the element e) for each e ∈ E , without changing the multiplication table. Then the groupoid will still satisfy (5.4) for these new sets.
for all e, f ∈ E and a ∈ N e , b ∈ N f , we have a 2 = e and (ab)c = e � c = e � = ab, a(bc) = af � = e = a 2 .

Theorem 5.7 A groupoid belongs to the variety L 4 if and only if there is a partition N e ∶ e ∈ E as in Notation 5.2, such that
Proof First assume that ∈ L 4 and recall that L 4 is defined by (xy)z ≈ xz and x(yz) ≈ x 2 . Let us consider the partition described in Remark 5.3. If a ∈ N e and b ∈ N f , then (ab) 2 = (ab)(ab) 1.
= a 2 = e , which means that ab ∈ S e . We also have ab = e 2 = e . This proves that (5.5) holds.
Conversely, suppose that N e ∶ e ∈ E is a partition of A as in Notation 5.2, such that (5.5) is satisfied. Let us compute (ab)c and a(bc) for a ∈ N e , b ∈ N f , c ∈ N g . From (5.5) we see that ab ∈ S e and bc ∈ S f . Therefore, using (5.5) again, we obtain This shows that satisfies (xy)z ≈ xz and x(yz) ≈ x 2 , hence ∈ L 4 . ◻

Theorem 5.8 A groupoid belongs to the variety L 5 if and only if there is a partition N e ∶ e ∈ E as in Notation 5.2, such that
Proof First assume that ∈ L 5 and recall that L 5 is defined by (xy)z ≈ xy and x(yz) ≈ xy . This time our partition will be different from that of Remark 5.3. We introduce an equivalence relation on A: let us write a ∼ b if the right multiplications by a and b coincide, i.e., if ca = cb for all c ∈ A (in other words, the columns of a and b in the multiplication table are the same). Since ca 2.
= c(aa) for every c ∈ A , we have a ∼ a 2 . As noted in Remark 5.3, a 2 is idempotent: (aa)(aa) 1.
= aa . Thus every equivalence class with respect to ∼ contains at least one idempotent element. Therefore, we can choose a complete set of representatives E consisting of idempotent elements. Let N e denote the equivalence class of e ∈ E , and let S e be the set of factorizable elements in N e , as before.
If a ∈ N e and b ∈ N f , then ab = af , as b ∼ f . Moreover, a ∼ e implies that for every c ∈ A , we have ce = ca 2.
= c(af ) , hence e ∼ af , which means that af ∈ S e . If a = a 1 a 2 ∈ S e , then ab = af = (a 1 a 2 )f 1.
= a 1 a 2 = a . This proves that (5.6) holds. (5.5) for all e, f ∈ E and a ∈ N e , b ∈ N f , we have a 2 = e and  for all e, f ∈ E and a ∈ N e , b ∈ N f , we have

Lazy groupoids
Conversely, suppose that N e ∶ e ∈ E is a partition of A as in Notation 5.2, such that (5.6) is satisfied. Let us compute (ab)c and a(bc) for a ∈ N e , b ∈ N f , c ∈ N g . From (5.6) we see that ab ∈ S e and bc ∈ S f . Therefore, using (5.6) again, we obtain This shows that satisfies (xy)z ≈ x(yz) ≈ xy , hence ∈ L 5 . ◻ Theorem 5.9 A groupoid belongs to the variety L 6 if and only if there is a partition N e ∶ e ∈ E as in Notation 5.2, such that Proof First assume that ∈ L 6 and recall that L 6 is defined by (xy)z ≈ xz and x(yz) ≈ xy . We use again the partition described in Remark 5.3. If a ∈ N e and b ∈ N f , then (ab) 2 = (ab)(ab) = a 2 = e , therefore ab ∈ S e . On the other hand, we can express ab as ab 1.
= (aa)(bb) = ef , which proves that (5.7) holds. Conversely, suppose that N e ∶ e ∈ E is a partition of A as in Notation 5.2, such that (5.7) is satisfied. Let us compute (ab)c and a(bc) for a ∈ N e , b ∈ N f , c ∈ N g . From (5.7) we see that ab ∈ S e and bc ∈ S f . Therefore, using (5.7) again, we obtain This shows that satisfies (xy)z ≈ xz and x(yz) ≈ xy , hence ∈ L 6 . Funding Open access funding provided by University of Szeged.
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(5.7) for all e, f ∈ E and a ∈ N e , b ∈ N f , we have ab = ef ∈ S e .
(ab)c = eg = ac, a(bc) = ef = ab.   Table 4 Summary of the proof of Lemma 3.4

Appendix 1: Detailed case analyses for the proofs of some lemmas
Lazy groupoids ≈ yz xy does not depend on x