Search Results (8)

If M is an algebra in a semidegenerate congruence-modular variety $\mathcal{V}$, then the set $Con(M)$ of congruences of M is an integral complete l-groupoid (= $icl$-groupoid). For any morphism $f:M\to N$ of $\mathcal{V}$, consider the map $f^{•}:Con(M)\to Con(N)$, where, for each congruence $\epsilon$ of M, $f^{•}(\epsilon )$ is the congruence of N generated by $f(\epsilon )$. Then, $f^{•}$ is a semimorphism of $icl$-groupoids, i.e., it preserves the arbitrary joins and the top congruences. The neo-commutative $icl$-groupoids were introduced recently by the author as an abstraction of the lattices of congruences of Kaplansky neo-commutative rings. In this paper, we define the admissible semimorphisms of $icl$-groupoids. The basic construction of the paper is a covariant functor defined by the following: $(1)$ to each semiprime and neo-commutative $icl$-groupoid A, we assign a coherent frame $R(A)$ of radical elements of A; and $(2)$ to an admissible semimorphism of $icl$-groupoids $u:A\to B$, we assign a coherent frame morphism $u^{\rho }:R(A)\to R(B)$. By means of this functor, we transfer a significant amount of results from coherent frames and coherent frame morphisms to the neo-commutative $icl$-groupoids and their admissible semimorphisms. We study the m-prime spectra of neo-commutative $icl$-groupoids and the going-down property of admissible semimorphisms. Using some transfer properties, we characterize some classes of admissible semimorphisms of $icl$-groupoids: Baer and weak-Baer semimorphisms, quasi r-semimorphisms, quasi $r^{*}$-semimorphisms, quasi rigid semimorphisms, etc. Full article

Let S be a 2-torsion free semiprime ring and U be a noncentral square-closed Lie ideal of $S.$ An additive mapping ℏ on S is defined as a homoderivation if $\hslash \left(ab\right)=\hslash \left(a\right)\hslash \left(b\right)+\hslash \left(a\right)b+a\hslash \left(a\right)$ for all $a,b\in S.$ In the present paper, we shall prove that ℏ is a commuting map on U if any one of the following holds: (i)$\hslash \left({\tilde{a}}_1{\tilde{a}}_2\right)+{\tilde{a}}_1{\tilde{a}}_2\in Z,$(ii)$\hslash \left({\tilde{a}}_1{\tilde{a}}_2\right)-{\tilde{a}}_1{\tilde{a}}_2\in Z,$(iii)$\hslash \left({\tilde{a}}_1\circ {\tilde{a}}_2\right)=0,$(iv)$\hslash \left({\tilde{a}}_1\circ {\tilde{a}}_2\right)=\left[{\tilde{a}}_1,{\tilde{a}}_2\right],$(v)$\hslash \left(\left[{\tilde{a}}_1,{\tilde{a}}_2\right]\right)=0,$(vi)$\hslash \left(\left[{\tilde{a}}_1,{\tilde{a}}_2\right]\right)=$ $({\tilde{a}}_1\circ {\tilde{a}}_2),$(vii)${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\pm {\tilde{a}}_1{\tilde{a}}_2\in Z,$(viii)${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\pm {\tilde{a}}_2{\tilde{a}}_1=0,$(ix)${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\pm {\tilde{a}}_1\circ {\tilde{a}}_2=0,$(x)$[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_2]\pm {\tilde{a}}_1{\tilde{a}}_2=0,$(xi)$[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_2]\pm {\tilde{a}}_2{\tilde{a}}_1=0,$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in U,$ where ℏ is a homoderivation on $S.$ Full article

This paper focuses on examining a new type of n-additive map called the symmetric reverse n-derivation. As implied by its name, it combines the ideas of n-additive maps and reverse derivations, with a 1-reverse derivation being the ordinary reverse derivation. We explore several findings that expand our knowledge of these maps, particularly their presence in semiprime rings and the way rings respond to specific functional identities involving elements of ideals. Also, we provide examples to help clarify the concept of symmetric reverse n-derivations. This study aims to deepen our understanding of these symmetric maps and their properties within mathematical structures. Full article

This paper investigates the relationship between the commutativity of rings and the properties of their multiplicative generalized derivations. Let $\mathcal{F}$ be a ring with a semiprime ideal $\Pi$. A map $\varphi :\mathcal{F}\to \mathcal{F}$ is classified as a multiplicative generalized derivation if there exists a map $\sigma :\mathcal{F}\to \mathcal{F}$ such that $\varphi \left(xy\right)=\varphi \left(x\right)y+x\sigma \left(y\right)$ for all $x,y\in \mathcal{F}$. This study focuses on semiprime ideals $\Pi$ that admit multiplicative generalized derivations $\varphi$ and G that satisfy certain differential identities within $\mathcal{F}$. By examining these conditions, the paper aims to provide new insights into the structural aspects of rings, particularly their commutativity in relation to the behavior of such derivations. Full article

This paper examines the commutativity of the quotient ring $\mathcal{F}/\mathrm{Y}$ by utilizing specific differential identities in a general ring $\mathcal{F}$ that contains a semiprime ideal $\mathrm{Y}$. This study particularly focuses on the role of a multiplicative generalized semiderivation $\psi$, which is associated with a map $\theta$, in determining the commutative nature of the quotient ring. Full article

We introduce the notions of meet, semi-prime, and prime weak closure operations. Using homomorphism of BCK-algebras $\phi :X\to Y$ , we show that every epimorphic image of a non-zeromeet element is also non-zeromeet and, for mapping $c{l}_Y:\mathcal{I}\left(Y\right)\to \mathcal{I}\left(Y\right)$ , we define a map $c{l}_Y^{\leftarrow }$ on $\mathcal{I}\left(X\right)$ by $A\mapsto {\phi }^{-1}\left(\phi {\left(A\right)}^{c{l}_Y}\right).$ We prove that, if “ $c{l}_Y$ ” is a weak closure operation (respectively, semi-prime and meet) on $\mathcal{I}\left(Y\right)$ , then so is “ $c{l}_Y^{\leftarrow }$ ” on $\mathcal{I}\left(X\right)$ . In addition, for mapping $c{l}_X:\mathcal{I}\left(X\right)\to \mathcal{I}\left(X\right)$ , we define a map $c{l}_X^{\to }$ on $\mathcal{I}\left(Y\right)$ as follows: $B\mapsto \phi \left({\phi }^{-1}{\left(B\right)}^{c{l}_X}\right).$ We show that, if “ $c{l}_X$ ” is a weak closure operation (respectively, semi-prime and meet) on $\mathcal{I}\left(X\right)$ , then so is “ $c{l}_X^{\to }$ ” on $\mathcal{I}\left(Y\right)$ . Full article

Prime numbers are routinely used in a variety of applications, e.g., cryptography and hashing. A prime number can only be divided by one and the number itself. A semi-prime number is a product of two or more prime numbers (e.g., 5 × 3 = 15) and can only be formed by these numbers (e.g., 3 and 5). Exploiting this mathematical property allows schema-free encoding of geographical data in nominal or ordinal measurement scales for thematic maps. Schema-free encoding becomes increasingly important in the context of data variety. In this paper, I investigate the encoding of categorical thematic map data using prime numbers instead of a sequence of all natural numbers (1, 2, 3, 4, ..., n) as the category identifier. When prime numbers are multiplied, the result as a single value contains the information of more than one location category. I demonstrate how this encoding can be used on three use-cases, (1) a hierarchical legend for one theme (CORINE land use/land cover), (2) a combination of multiple topics in one theme (Köppen-Geiger climate classification), and (3) spatially overlapping regions (tree species distribution). Other applications in the field of geocomputation in general can also benefit from schema-free approaches with dynamic instead of handcrafted encoding of geodata. Full article

Detailed information from global remote sensing has greatly advanced ourunderstanding of Earth as a system in general and of agricultural processes in particular.Vegetation monitoring with global remote sensing systems over long time periods iscritical to gain a better understanding of processes related to agricultural change over longtime periods. This specifically relates to sub-humid to semi-arid ecosystems, whereagricultural change in grazing lands can only be detected based on long time series. Byintegrating data from different sensors it is theoretically possible to construct NDVI timeseries back to the early 1980s. However, such integration is hampered by uncertainties inthe comparability between different sensor products. To be able to rely on vegetationtrends derived from integrated time series it is therefore crucial to investigate whether vegetation trends derived from NDVI and phenological parameters are consistent acrossproducts. In this paper we analyzed several indicators of vegetation change for a range ofagricultural systems in Inner Mongolia, China, and compared the results across differentsatellite archives. Specifically, we compared two of the prime NDVI archives—AVHRR Global Inventory Modeling and Mapping Studies (GIMMS) and SPOT Vegetation (VGT)NDVI. Because a true accuracy assessment of long time series is not possible, we furthercompared SPOT VGT NDVI with NDVI from MODIS Terra as a benchmark. We foundhigh similarities in interannual trends, and also in trends of the seasonal amplitude andintegral between SPOT VGT and MODIS Terra (r > 0.9). However, we observedconsiderable disagreements in NDVI-derived trends between AVHRR GIMMS and SPOTVGT. We detected similar discrepancies for trends based on phenological parameters, suchas amplitude and integral of NDVI curves corresponding to seasonal vegetation cycles.Inconsistencies were partially related to land cover and vegetation density. Differentpre-processing schemes and the coarser spatial resolution of AVHRR GIMMS introducedfurther uncertainties. Our results corroborate findings from other studies that vegetationtrends derived from AVHRR GIMMS data not always reflect true vegetation changes. Amore thorough understanding of the factors introducing uncertainties in AVHRR GIMMStime series is needed, and we caution against using AVHRR GIMMS data in regionalstudies without applying regional sensitivity analyses. Full article