Abstract
Abstract. We find an independent base for three-variable equations of posets.
Introduction
For a poset (P, ≤) we can define multiplication on P by xy = x if x ≤ y, and xy = y otherwise. The groupoid (P, ·) will be called the poset groupoid of (P, ≤). Since the correspondence between posets and poset groupoids is one-to-one, posets will be sometimes identified with their poset groupoids.
Introduction
More generally, one can identify an arbitrary directed graph with loops (a set with a reflexive binary relation) with a groupoid defined in a similar way, and then employ methods of universal algebra to investigate various interesting classes of directed graphs. This approach was started in [5] and continued by various authors in the case of the class of tournaments. The variety generated by tournaments has been investigated in [1] and [2] . In [3] , we investigate in a similar way the variety generated by equivalence relations. The variety generated by posets (partially ordered sets) seems to be the most natural and interesting candidate in this respect. In the present paper we start to investigate this variety.
Introduction
We denote by P the variety generated by posets (or poset groupoids). For any n ≥ 1, let P n denote the variety generated by all n-element posets, and let P n denote the variety determined by the at most n-variable equations of posets. So, P n ⊆ P n+1 ⊆ P ⊆ P n+1 ⊆ P n for all n. It is easy to see that the free groupoid on n generators in P is a free groupoid on n generators in P n , as well as in P n . The aim of this paper is to describe the free groupoid on three generators in P, and to find an independent base for the equations of the variety P 3 . We leave as an open problem the question whether the variety P is finitely based.
A base for the equations of P 3
It is easy to check that every poset groupoid satisfies the following five equations:
A base for the equations of P 3
(
A base for the equations of P 3
While working on this paper the author was partially supported by the Grant Agency of the Czech Republic, grant #201/99/0263 and by the institutional grant MSM113200007.
A base for the equations of P 3
The following two-variable equations are consequences of (1)- (5):
A base for the equations of P 3
For a groupoid A ∈ P 3 put x ≤ y if and only if xy = x. We are now able to show that this relation is an ordering on P . Reflexivity is clear. If x ≤ y and y ≤ z, then xz = (6) x · xz = x(xy · z) = (4) x · yz = xy = x, i.e., x ≤ z. If x ≤ y and y ≤ z, then x = xy = (2) yx · y = yy = (1) y.
A base for the equations of P 3
We continue with a list of three-variable consequences of (1)- (5):
A base for the equations of P 3
use the transitivity of ≤. (14) xy ·zy = x·zy. Proof: xy ·zy = (5) ((x·zy)y)·zy = (13) (x·zy)·zy = (3) x · zy.
A base for the equations of P 3
41) (xy·z)(zx·y) = z·xy. Proof: (xy·z)(zx·y) = (10) (xy·z)(zx·xy) = (11) z(zx · xy) = (10) z(zx · y) = (4) z · xy. (42) (xy·z)(yz ·x) = yz ·x. Proof: (xy·z)(yz ·x) = (10) (xy·yz)(yz ·x)
A base for the equations of P 3
44) (xy ·z)(x·yz) = x·yz. Proof: (xy ·z)(x·yz) = (4) (xy ·z)·x(xy ·z) = (7)
A base for the equations of P 3
x(xy · z) = (4) x · yz. (45) (xy · z)(x · zy) = x · zy. Proof: (xy · z)(x · zy) = (20) (xy · z)(xy · (xz · y)) = (9) z(xy · (xz · y)) = (20) z(x · zy) = (16) x · zy. (46) (xy · z)(y · xz) = y · xz. Proof: (xy · z)(y · xz) = (9) (xy · z)(xy · xz) = (9) z(xy · xz) = (9) z(y · xz) = (18) y · xz. (47) (xy ·z)(y ·zx) = y ·zx. Proof: (xy ·z)(y ·zx) = (11) (xy ·z)(xy ·zx) = (9) z(xy · zx) = (11) z(y · zx) = (16) y · zx. (48) (xy·z)(z·yx) = z·yx. Proof: (xy·z)(z·yx) = (10) (xy·z)·yx = (29) z·yx. (49) (x·yz)(xy ·z) = x·yz. Proof: (x·yz)(xy ·z) = (4) x(xy ·z)·(xy ·z) = (3) x(xy · z)
A base for the equations of P 3
53) (x·yz)(zy·x) = z·yx. Proof: (x·yz)(zy·x) = (11) yz·(zy·x) = (19) z·yx. (54) (x·yz)(x·zy) = x·zy. Proof: (x·yz)(x·zy) = (9) yz·(x·zy) = (28) x·zy.
A base for the equations of P 3
57) (x·yz)(z·xy) = z·xy. Proof: (x·yz)(z·xy) = (4) x(xy·z)·(z·xy) = (38) xz · xy = (9) z · xy. (58) (x·yz)(z·yx) = z·yx. Proof: (x·yz)(z·yx) = (4) x(xy·z)·(z·yx) = (43) x(xy ·z)·(xy ·z)(zy ·x) = (10) x(xy ·z)·(zy ·x) = (4) (x·yz)(zy ·x)
A base for the equations of P 3
Theorem 1. The equations (1)- (5) constitute a base for the equational theory of P 3 . The free groupoid on three generators x, y, z in P has 21 elements q r s t u f g l f t u f g p l p k l t n u p q r s t u g g m f m u f g p q p k q m n u p q r s t u h n i h t u r n h i j r l t n u p q r s t u i o i h t o r s h i j r l t s o p q r s t u j n l j t u r n j l j r l t n u p q r s t u k n l k t u k n p l p k l t n u p q r s t u l o l j t o r s j l j r l t s o p q r s t u m o m j m o r s j q j r q m s o p q r s t u n n m k m u k n p q p k q m n u p q r s t u o o m k m o k s p q p k q m s o p q r s t u p n l p t u r n p l p r l t n u p q r s t u q o q j t o r s j q j r q t s o p q r s t u r n l r t u r n p l p r l t n u p q r s t u s s m k m u k s p q p k q m s u p q r s t u t o t j t o r s j q j r q t s o p q r s t u u u m k m u k s p q p k q m s u p q r s t u Proof. Using the above listed 58 equations, it is easy to build the multiplication table of the free groupoid on three generators x, y, z. In order to see that the 21 elements are pairwise different, it is possible to check that the 21 terms behave differently on three-element posets.
A base for the equations of P 3
3. Single-exception differences from poset groupoids Groupoids that do not differ much from poset groupoids can be conveniently described by a list of the exceptions ab = c from the poset groupoid multiplication. Given a poset P and a triple of elements a, b, c of P , we denote by P [ab = c] the groupoid with multiplication defined by
A base for the equations of P 3
x if x ≤ y and (x, y) = (a, b), y if x y and (x, y) = (a, b).
A base for the equations of P 3
Similarly as in the case of tournaments ( [1] and [2] ), these single-exception non-poset groupoids may turn out to be crucial for the investigation of the equational theory of poset groupoids. (1)- (5) but (5) .

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