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.
This paper analyses the so-called Chapbooks that were being written or translated on the dawn of the 19th century. The authors tried to educate the ignorant peasants, the targeted readers, through their fetching stories. The work shows facts and deeds that were presented as "right" and "wise". First of all it presents the factual public enlightenment, more specifically the altering appreciation of time. Next, there is an analysis of the way the authors were maintaining the cogency of their work; the paper discusses whether the narrative style of writing is compatible with the didactic intention, and the characteristics of the "rational order of explanation"., Barbora Matiášová., and Obsahuje bibliografické odkazy
Let $S$ be a regular semigroup and $E(S)$ be the set of its idempotents. We call the sets $S(e,f)f$ and $eS(e,f)$ one-sided sandwich sets and characterize them abstractly where $e,f \in E(S)$. For $a, a^{\prime } \in S$ such that $a=aa^{\prime }a$, $a^{\prime }=a^{\prime }aa^{\prime }$, we call $S(a)=S(a^{\prime }a, aa^{\prime })$ the sandwich set of $a$. We characterize regular semigroups $S$ in which all $S(e,f)$ (or all $S(a))$ are right zero semigroups (respectively are trivial) in several ways including weak versions of compatibility of the natural order. For every $a \in S$, we also define $E(a)$ as the set of all idempotets $e$ such that, for any congruence $\rho $ on $S$, $a \rho a^2$ implies that $a \rho e$. We study the restrictions on $S$ in order that $S(a)$ or $E(a)\cap D_{a^2}$ be trivial. For $\mathcal F \in \lbrace \mathcal S, \mathcal E\rbrace $, we define $\mathcal F$ on $S$ by $a \mathrel {\mathcal F}b$ if $F(a) \cap F (b)\ne \emptyset $. We establish for which $S$ are $\mathcal S$ or $\mathcal E$ congruences.