We introduce the concept of vague ideals in a distributive implication groupoid and investigate their properties. The vague ideals of a distributive implication groupoid are also characterized.

Using the notion of a vaguely defined object, we systematize and unify different existing approaches to vagueness and its mathematical representations, including fuzzy sets and derived concepts. Moreover, a new, approximative approach to vaguely defined objects will be introduced and investigated.

We prove that the set of asymptotic critical values of a $C^1$ function definable in an o-minimal structure is finite, even if the structure is not polynomially bounded. As a consequence, the function is a locally trivial fibration over the complement of this set.

The validation set of a formula in a fuzzy logic is the set of all truth values which this formula may achieve. We summarize characterizations of validation sets of $S$-fuzzy logics and extend them to the case of $R$-fuzzy logics.

We enlarge the problem of valuations of triads on so called lines. A line in an $e$-structure $𝔸=\langle A,F,E\rangle$ (it means that $\langle A,F\rangle$ is a semigroup and $E$ is an automorphism or an antiautomorphism on $\langle A,F\rangle$ such that $E\circ E=\mathrm{𝐈𝐝}\upharpoonright A$) is, generally, a sequence $𝔸\upharpoonright B$, $𝔸\upharpoonright U_c$, $c\in \mathrm{𝐅𝐙}$ (where $\mathrm{𝐅𝐙}$ is the class of finite integers) of substructures of $𝔸$ such that $B\subseteq U_c\subseteq U_d$ holds for each $c\le d$. We denote this line as $𝔸{(U_c,B)}_{c\in \mathrm{𝐅𝐙}}$ and we say that a mapping $H$ is a valuation of the line $𝔸{(U_c,B)}_{c\in \mathrm{𝐅𝐙}}$ in a line $\widehat{𝔸}{({\widehat{U}}_c,\widehat{B})}_{c\in \mathrm{𝐅𝐙}}$ if it is, for each $c\in \mathrm{𝐅𝐙}$, a valuation of the triad $𝔸(U_c,B)$ in $\widehat{𝔸}({\widehat{U}}_c,\widehat{B})$. Some theorems on an existence of...

A characterization of locally finite congruence modular varieties with the number of at most k-generated models being bounded from above by a polynomial in k is given. These are exactly the varieties polynomially equivalent to the varieties of unitary modules over a finite ring of finite representation type.

The Veblen hierarchy is an extension of the construction of epsilon numbers (fixpoints of the exponential map: ωε = ε). It is a collection φα of the Veblen Functions where φ0(β) = ωβ and φ1(β) = εβ. The sequence of fixpoints of φ1 function form φ2, etc. For a limit non empty ordinal λ the function φλ is the sequence of common fixpoints of all functions φα where α < λ.The Mizar formalization of the concept cannot be done directly as the Veblen functions are classes (not (small) sets). It is done...

We consider the question when a set in a vector space over the rationals, with no differences occurring more than twice, is the union of countably many sets, none containing a difference twice. The answer is “yes” if the set is of size at most ${\aleph }_2$, “not” if the set is allowed to be of size ${\left(2^{2^{{\aleph }_0}}\right)}^{+}$. It is consistent that the continuum is large, but the statement still holds for every set smaller than continuum.