Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

Si considera il calcolo modale interpretato $ℳ{𝒞}^{\nu }$, che è basato su un sistema di tipi con infiniti livelli, contiene descrizioni, ed è dotato di una semantica di tipo generale - v. [2], o [3], o [4], o [5]. In modo semplice e naturale si introducono in $ℳ{𝒞}^{\nu }$ operatori vincolanti variabili, di tipo generale. Per teorie basate sul calcolo logico risultante $ℳ{𝒞}^{\nu }$ vale un teorema di completezza, che si dimostra in modo immediato sulla base dell'estensione del teorema parziale di completezza stabilito in [11], fatta...

We deal with unbounded dually residuated lattices that generalize pseudo $MV$-algebras in such a way that every principal order-ideal is a pseudo $MV$-algebra. We describe the connections of these generalized pseudo $MV$-algebras to generalized pseudo effect algebras, which allows us to represent every generalized pseudo $MV$-algebra $A$ by means of the positive cone of a suitable $\ell$-group $G_A$. We prove that the lattice of all (normal) ideals of $A$ and the lattice of all (normal) convex $\ell$-subgroups of $G_A$ are isomorphic....

We introduce an analog to the notion of Polish space for spaces of weight ≤ κ, where κ is an uncountable regular cardinal such that ${\kappa }^{<\kappa }=\kappa$. Specifically, we consider spaces in which player II has a winning strategy in a variant of the strong Choquet game which runs for κ many rounds. After discussing the basic theory of these games and spaces, we prove that there is a surjectively universal such space and that there are exactly $2^{\kappa }$ many such spaces up to homeomorphism. We also establish a Kuratowski-like...

We analyze the existence of fuzzy sets of a universe that are convex with respect to certain particular classes of fusion operators that merge two fuzzy sets. In addition, we study aggregation operators that preserve various classes of generalized convexity on fuzzy sets. We focus our study on fuzzy subsets of the real line, so that given a mapping $F:[0,1]\times [0,1]\to [0,1]$, a fuzzy subset, say $X$, of the real line is said to be $F$-convex if for any $x,y,z\in ℝ$ such that $x\le y\le z$, it holds that ${\mu }_X\left(y\right)\ge F({\mu }_X\left(x\right),{\mu }_X\left(z\right))$, where ${\mu }_X:ℝ\to [0,1]$ stands here for the membership function...

An algebra $𝒜=(A,F)$ is subregular alias regular with respect to a unary term function $g$ if for each $\Theta ,\Phi \in \text{Con}𝒜$ we have $\Theta =\Phi$ whenever ${\left[g\left(a\right)\right]}_{\Theta }={\left[g\left(a\right)\right]}_{\Phi }$ for each $a\in A$. We borrow the concept of a deductive system from logic to modify it for subregular algebras. Using it we show that a subset $C\subseteq A$ is a class of some congruence on $\Theta$ containing $g\left(a\right)$ if and only if $C$ is this generalized deductive system. This method is efficient (needs a finite number of steps).

An R-algebra A is called an E(R)-algebra if the canonical homomorphism from A to the endomorphism algebra $En{d}_{R}A$ of the R-module ${}_{R}A$, taking any a ∈ A to the right multiplication $a_r\in En{d}_{R}A$ by a, is an isomorphism of algebras. In this case ${}_{R}A$ is called an E(R)-module. There is a proper class of examples constructed in [4]. E(R)-algebras arise naturally in various topics of algebra. So it is not surprising that they were investigated thoroughly in the last decades; see [3, 5, 7, 8, 10, 13, 14, 15, 18, 19]. Despite...

We study unbounded versions of effect algebras. We show a necessary and sufficient condition, when lattice operations of a such generalized effect algebra $P$ are inherited under its embeding as a proper ideal with a special property and closed under the effect sum into an effect algebra. Further we introduce conditions for a generalized homogeneous, prelattice or MV-effect effect algebras. We prove that every prelattice generalized effect algebra $P$ is a union of generalized MV-effect algebras and...