On elementary equivalence of some unoids and unoids of their subsets

В данной работе исследуются свойства уноидов, которые содержат единственную разнозначную функцию. Устанавливаются необходимые и достаточные условия для того, чтобы два таких уноида были элементарно эквивалентными. С помощью этого результата доказываются необходимые и достаточные условия для того, чтобы уноид всех подмножеств уноида $\gA$ был элементарно эквивалентен исходному уноиду $\gA$. In this paper we study the properties of unoids which contain a single injective function. Necessary and sufficient conditions are established for two such unoids to be elementarily equivalent. From this result we obtain necessary and sufficient conditions for the unoid of all subsets of unoid $\gA$ to be elementarily equivalent to the original unoid $\gA$.

Condensed groups in product varieties

A finitely generated group 𝐺 is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points. We prove that every product variety U ⁢ V \mathcal{UV} , where 𝒰 (respectively, 𝒱) is a non-abelian (respectively, a non-locally finite) variety, contains a condensed group. In particular, there exist condensed groups of finite exponent. As an application, we obtain some results on the structure of the isomorphism and elementary equivalence relations on the set of finitely generated groups in U ⁢ V \mathcal{UV} .

Fraïssé–Hintikka theorem in institutions

We generalize the characterization of elementary equivalence by Ehrenfeucht–Fraïssé games to arbitrary institutions whose sentences are finitary. These include many-sorted first-order logic, higher-order logic with types, as well as a number of other logics arising in connection to specification languages. The gain for the classical case is that the characterization is proved directly for all signatures, including infinite ones.