### A common generalization of permutability and 0-permutability

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The contribution is devoted to the question of the interchange of the construction of a quasiorder hypergroup from a quasiordered set and the factorization.

The categorical concept of a theory for algebras of a given type was foundet by Lawvere in 1963 (see [8]). Hoehnke extended this concept to partial heterogenous algebras in 1976 (see [5]). A partial theory is a dhts-category such that the object class forms a free algebra of type (2,0,0) freely generated by a nonempty set J in the variety determined by the identities ox ≈ o and xo ≈ o, where o and i are the elements selected by the 0-ary operation symbols. If the object class of a dhts-category...

Given a groupoid $\langle G,\u2606\rangle $, and $k\ge 3$, we say that $G$ is antiassociative if an only if for all ${x}_{1},{x}_{2},{x}_{3}\in G$, $({x}_{1}\u2606{x}_{2})\u2606{x}_{3}$ and ${x}_{1}\u2606({x}_{2}\u2606{x}_{3})$ are never equal. Generalizing this, $\langle G,\u2606\rangle $ is $k$-antiassociative if and only if for all ${x}_{1},{x}_{2},...,{x}_{k}\in G$, any two distinct expressions made by putting parentheses in ${x}_{1}\u2606{x}_{2}\u2606{x}_{3}\u2606\cdots \u2606{x}_{k}$ are never equal. We prove that for every $k\ge 3$, there exist finite groupoids that are $k$-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.

Boolean matrices, the incidence matrices of a graph, are known not to be the (universal) matrices of a Boolean algebra. Here, we also show that their usual composition cannot make them the matrices of any algebra. Yet, later on, we "show" that it can. This seeming paradox comes from the hidden intrusion of a widespread set-theoretical (mis) definition and notation and denies its harmlessness. A minor modification of this standard definition might fix it.

The aim of this paper is to define and study cardinal (direct) and ordinal operations of addition, multiplication, and exponentiation for $n$-ary relational systems. $n$-ary ordered sets are defined as special $n$-ary relational systems by means of properties that seem to suitably generalize reflexivity, antisymmetry, and transitivity from the case of $n=2$ or 3. The class of $n$-ary ordered sets is then closed under the cardinal and ordinal operations.

It is well-known that the composition of two functors between categories yields a functor again, whenever it exists. The same is true for functors which preserve in a certain sense the structure of symmetric monoidal categories. Considering small symmetric monoidal categories with an additional structure as objects and the structure preserving functors between them as morphisms one obtains different kinds of functor categories, which are even dt-symmetric categories.