We generalize Cauchy's Lemma on finite groups to specific classes of finite hypergroups.
We generalize Cauchy’s Lemma on finite groups to specific classes of finite hypergroups.
The set of Boolean functions is generated by the two operations of conjunction or multiplication modulo 2 ( ∧ ) and negation (−). Is it possible to have a similar basis for the set of finitary operations on a finite set with more than three elements? A clone on a finite set is a set of finitary operations on that set containing all its projections and stable under composition. It is known that any clone of operations on a finite set can be expressed as a set of operations preserving some relations on this set. It is also known that the set of clones on a finite set is a complete lattice. The maximal elements of this lattice were completely described by Rosenberg in 1965. In this paper, we determine the clones of the form Pol σ∩ Pol λ located directly below Pol λ , where λ is a fixed regular relation, σ is a binary relation to be characterized, and Pol λ is the set of functions preserving λ . More precisely, we show that Pol σ∩ Pol λ is located directly below Pol λ if and only if σ is a non-trivial equivalence relation and λ is σ -closed (i.e., λ contains the product of σ -classes); σ is a non-trivial equivalence relation and for all x,y ∈ E_k , [x]_σ∩ [y]_ε∅ (with [x]_σ being the σ -class of x and ε is the equivalence relation characterizing λ ); σ is a prime permutation and λ is θ _σ -closed; or σ is a binary central relation such that for all x∈ E_k , C_σ∩ [x]_ε∅ (with C_σ being the set of central elements of σ ).
We consider two questions asking whether a recently constructed effect algebra is smallest in two classes determined by its sharp part. The first class consists of non-homogeneous effect algebras whose sharp part is the four-element Boolean algebra. The second consists of effect algebras whose Boolean sharp part is different from the center. We show that the original eleven-element example is not smallest: the first question is answered by an eight-element algebra, while the second has a five-element answer. We also record the homogeneous and non-homogeneous readings of the second question.
In this paper, we employ the concept of a ζ -ideal 𝒥 and introduce a quotient frame M(X,𝒥) , where X is a topological space. We prove that for a metric space X, this frame is metrizable. Moreover, a necessary and sufficient condition for spatiality of frame M(X,𝒥) will be given. Finally, we apply this general case to the metric space ℝ^n and construct frames which are metrizable but not spatial. More precisely, these frames are derived from the metric space ℝ^n which are actually the quotients of frame 𝒪(ℝ^n) , the set of open subsets of ℝ^n . The technique of constructing these frames, which are denoted by M_i(X, S) , i=0,1,2 , intuitively, is a process in which a given subset of points are removed.
