We introduce and investigate the concept of flexibility in a groupoid, and extend it to RM-algebras, and present several examples and these properties. We observe that condition flexibility is strong, and most known algebras of logic will be trivial with this condition. It is proved that every KL-algebra and BE-algebra are flexible, and every flexible CI-algebra is a BE- algebra. We present that the flexible groups are trivial. Moreover, we apply two conditions (I) and (H) in a groupoid and prove that in a RM-algebra, they will be equivalent.

It is known that even duals of a Banach algebra A with one of Arens products are Banach algebras, these products are natu-ral multiplications extending the one on A. But the essence of A ∗,is completely di, erent. By de, ning new products, we investigate some algebraic and spectral properties of odd duals of A. We will show re-lations between these products and Arens products, weak or weak-star continuity, commutativity and unit elements of these algebras. We also determine the spectrum and multiplier algebra for A ∗, , and we calculate the quasi-inverses, spectrum and spectral radius for elements of these kinds of algebras.

For a locally compact group G, let A(G) be the Fourier algebra and let A_M (G) be the completion of this algebra in its multiplier algebra. In this paper, we show that A(G) is an abstract Segal algebra in A_M (G). Also, a necessary and sufficient condition for equality of these two algebras is given. Then we prove that A_M (G) is an ideal in its second dual if and only if G is discrete. We show that if G is a discrete group, then A_M (G) is a BSE algebra if and only if G is M-weakly amenable. As a corollary, it is proven that A_M (F_2 ) is a BSE algebra while A(F_2 ) is not. Finally, we examine our results for the Lebasque-Fourier algebra and also give a completely new proof for equality of the character space of A(G) and A_M (G).

Usuli experts' theory of conceptual implication, on one hand, and the linguistic philosophers' theory of semantical implication, on the other, can be the subject of a comparative study. Theory of conceptual implication is the basis of the semantic approach. Conceptual implication is the transfer of mind from a term to its meaning regardless of the speaker's intention. Judicial implication is the transfer of mind from a term to the speaker's intended meaning. Semantics is closer to the theory of conceptual implication because semantics is the study of meaning independent of the speaker or author’s intention. Pragmatics is closer to theory of judicial implication because pragmatics is the study of meaning based on the speaker or author’s intention. Of all theories of semantical signification, the use theory of meaning is closer to the theory of conceptual implication because according to both theories, language and the use of terms are socially rule-governed activities. What's more, both theories claim that phrases and terms acquire their meaning through their multiple usages and functions in society.

We show that the radical bundle of an algebra bundle is a characteristic ideal bundle. Further we prove an algebra bundle is semisimple if and only if its derivation algebra bundle is either semisimple or zero.

Sound, with all its manifestations, presents itself as a form of energy with properties and capabilities that remain partially shrouded in mystery. The art of composing music, the artful assemblage and fusion of sounds, still lingers in a realm of obscurity, fraught with complexities. The techniques wielded by composers often stem from an intrinsic gift that defies replication or sharing. However, akin to other art forms, music can be deciphered through the lenses of mathematics and geometry. Through the utilization of linear algebra, a paramount mathematical discipline, a practical and efficient framework can be established for the comprehension and manipulation of musical compositions. Within this discourse, we delve into the essence of music and its fundamental principles, subsequently exploring the application of mathematical concepts; linear algebra in particular; in the analysis of music. The findings put forth in this manuscript hold the potential to enlighten researchers and enthusiasts within the realm of music and mathematical sciences, enhance comprehension and knowledge, refine and enhance analytical methodologies and musical procedures across various facets of the music industry, and foster a deeper bond between the realms of mathematics and artistic expression.