Given a groupoid hG, ⋆i, and k ≥ 3, we say that G is antiassociative if an only if for all x1, x2, x3 ∈ G, (x1 ⋆ x2) ⋆ x3 and x1 ⋆ (x2 ⋆ x3) are never equal. Generalizing this, hG, ⋆i is k-antiassociative if and only if for all x1, x2, . . . , xk ∈ G, any two distinct expressions made by putting parentheses in x1 ⋆ x2 ⋆ x3 ⋆ . . . ⋆ xk are never equal. We prove that for every k ≥ 3, there exist finite groupoids that are k-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.
The notion of Cayley color graphs of groups is generalized to inverse semigroups and groupoids. The set of partial automorphisms of the Cayley color graph of an inverse semigroup or a groupoid is isomorphic to the original inverse semigroup or groupoid. The groupoid of color permuting partial automorphisms of the Cayley color graph of a transitive groupoid is isomorphic to the original groupoid.
In a groupoid, consider arbitrarily parenthesized expressions on the k variables x0, x1, . . . xk−1 where each xi appears once and all variables appear in order of their indices. We call these expressions k-ary formal products, and denote the set containing all of them by F σ (k). If u, v ∈ F σ (k) are distinct, the statement that u and v are equal for all values of x0, x1, . . . xk−1 is a generalized associative law. Among other results, we show that many small groupoids are completely dissociative, meaning that no generalized associative law holds in them. These include the two groupoids on {0, 1} where the groupoid operation is implication and NAND, respectively.
The study of paramedial groupoids (with emphasis on the structure of simple paramedial groupoids) was initiated in [1] and continued in [2], [3] and [5]. The aim of the present paper is to give a full description of finite simple zeropotent paramedial groupoids (i.e., of finite simple paramedial groupoids of type (II)—see [2]). A reader is referred to [1], [2], [3] and [7] for notation and various prerequisites.
By a relational system we mean a couple (A, R) where A is a set and R is a binary relation on A, i.e. R ⊆ A × A. To every directed relational system A = (A, R) we assign a groupoid G(A) = (A, ·) on the same base set where xy = y if and only if (x, y) ∈ R. We characterize basic properties of R by means of identities satisfied by G(A) and show how homomorphisms between those groupoids are related to certain homomorphisms of relational systems.
Slim groupoids are groupoids satisfying $x(yz)\=xz$. We find all simple slim groupoids and all minimal varieties of slim groupoids. Every slim groupoid can be embedded into a subdirectly irreducible slim groupoid. The variety of slim groupoids has the finite embeddability property, so that the word problem is solvable. We introduce the notion of a strongly nonfinitely based slim groupoid (such groupoids are inherently nonfinitely based) and find all strongly nonfinitely based slim groupoids with at most four elements; up to isomorphism, there are just two such groupoids.
Idempotent slim groupoids are groupoids satisfying $xx\=x$ and $x(yz)\=xz$. We prove that the variety of idempotent slim groupoids has uncountably many subvarieties. We find a four-element, inherently nonfinitely based idempotent slim groupoid; the variety generated by this groupoid has only finitely many subvarieties. We investigate free objects in some varieties of idempotent slim groupoids determined by permutational equations.