We provide a generalization of Mundici's equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category commutative is equivalent to MV-monoidal algebras. Roughly speaking, structures we call are without unary operation $x \mapsto -x$. The primitive operations $+$, $\lor$, $\land$, $0$, $1$, $-1$. A prime example these $\mathbb{R}$, with obvious interpretation operation...

Abstract For a Galois extension $K/F$ with $\text {char}(K)\neq 2$ and $\mathrm {Gal}(K/F) \simeq \mathbb {Z}/2\mathbb {Z}\oplus {Z}$ , we determine the $\mathbb {F}_{2}[\mathrm {Gal}(K/F)]$ -module structure of $K^{\times }/K^{\times 2}$ . Although there are an infinite number (pairwise nonisomorphic) indecomposable {F}_{2}[\mathbb {Z}]$ -modules, our decomposition includes at most nine types....

Abstract We examine the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ of equivalence relations on $\omega $ under computable reducibility. when pairs degrees have a least upper bound. In particular, we show that sufficiently incomparable do not bound but some do, and characterize which with every finite relation. natural classes finite, light, dark are definable in . relation has co...

Let $\go{g}$ be a finite-dimensional semi-simple Lie algebra, $\go{h}$ a Cartan subalgebra of $\go{g}$, and $W$ its Weyl group. The group $W$ acts diagonally on $V:=\go{h}\oplus\go{h}^*$, as well as on $\mathbb{C}[V]$. The purpose of this article is to study the Poisson homology of the algebra of invariants $\mathbb{C}[V]^W$ endowed with the standard symplectic bracket. To begin with, we give g...

We give an explicit formula showing how the double Poisson algebra introduced in \cite{VdB} appears as a particular part of pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to natural inner form, solution Maurer-Cartan equation on $A\oplus A^*$. Specific this is described, which one-to-one correspondence structures. The result holds for any associative $A$ and emphasizes specia...

The purpose of this note is to describe when a general complex algebraic $^*$-algebra pre-$C^*$-normed, and investigate their structure the $^*$-algebras are Baer $^*$-rings in addition algebraicity. As main result we prove following theorem for $^*$-algebras: every kind can be decomposed as direct sum $M\oplus B$, where $M$ finite dimensional $B$ commutative $^*$-algebra. summand $^*$-isomorph...

In an interesting recent paper [1] (A. Acharya, Stress of a spatially uniform dislocation density field, J. Elasticity 137 (2019), 151--155), Acharya proved that the stress produced by field in body comprising nonlinear elastic material may fail to vanish under no loads. The class counterexamples constructed is essentially $2$-dimensional: it works with subgroup $\mathcal{O}(2) \oplus \langle{\...

Variability in the far ultraviolet (FUV) emission produced by stellar activity affects photochemistry and heating orbiting planetary atmospheres. We present a comprehensive analysis of FUV variability GJ 436, field-age, M2.5V star ($P_\mathrm{rot}\approx44$ d) orbited warm, Neptune-size planet ($M \approx 25\ M_\oplus$, $R 4.1\ R_\oplus$, $P_\mathrm{orb}\approx2.6$ d). Observations at three epo...

A multiplication on an Abelian group G is a homomorphism μ: ⊗ → G. An with it called ring the R. A. Beaumont and D. Lawver have formulated problem of studying semisimple groups. said to be if there exists associative it. Semisimple groups are described in class vector nonmeasurable It also shown that set I nonmeasurable, $$ G=\coprod_{i\in I}{A}_i reduced group, μ G, then determined by its rest...

All \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) sending linear \(3\)-forms \(H \in \Gamma^{l}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^\infty\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear \[C_H:\Gamma^l_E(TE \oplus T^*E)\times \Gamma^l_E(TE T^*E)\to T^*E)\] transforming pairs of sections \(TE T^*E \to E\) \( TE are completely described. The complete descriptions is given a...