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For example, a ring is a structure of the language $\{+,-,\times,0,1\}$, and a ring is not a structure of the language $\{+,-,\times,\cdot^{-1},0,1\}$ because the domain of the operation $\cdot^{-1}$ of the multiplicative inverse is not the whole. In general, it is not officially possible to introduce a function symbol into a partial function. In this paper, we consider ``a structure in a broad sense'' that allows a partial function as the interpretation of a function symbol, we give its semantics and a Hilbert-style formal deductive system, and we prove the completeness theorem. Regarding sequent calculus and natural deduction, it may not be difficult, but it is an unsolved problem. ------ Intuitionistic logic, Kripke model, Universal TM, Derivability condition. (This is the Japanese version.), Japanese version

Let S be a pre-Hilbert space. We study quasi-splitting subspaces of S and compare the class of such subspaces, denoted by Eq(S), with that of splitting subspaces E(S). In [D. Buhagiar, E. Chetcuti, Quasi splitting subspaces in a pre-Hilbert space, Math. Nachr. 280 (5–6) (2007) 479–484] it is proved that if S has a non-zero finite codimension in its completion, then Eq(S) = E(S). In the present paper it is shown that if S has a total orthonormal system, then Eq(S) = E(S) implies completeness of S. In view of this result, it is natural to study the problem of the existence of a total orthonormal system in a pre-Hilbert space. In particular, it is proved that if every algebraic complement of S in its completion is separable, then S has a total orthonormal system., peer-reviewed

In the present paper the idea of Wang [G.J. Wang, Theory of truth degrees of formulas in Łukasiewicz n-valued propositional logic and a limit theorem, Sci. China Inform. Sci. E 35(6) (2005) 561–569 (in Chinese)] is firstly extended to the n-valued R0-logic L n ∗ and the concept of truth degrees of formulas in L n ∗ is proposed. A limit theorem saying that the truth function τn induced by truth degrees converges to the integrated truth function τ when n converges to infinity is obtained. This theorem builds a bridge between discrete valued R0-logic and continuous valued R0-logic. Secondly, based on deduction theorem, completeness theorem and the concept of truth degrees of formulas in L n ∗ , the concept of consistency degrees of theories is given. It is proved that a theory Γ over L n ∗ is a useless theory(i.e., the deductions of Γ are all tautologies) iff the consistency degree consistn(Γ) of Γ is equal to 1, Γ is consistent iff 1 2 ⩽ consist n ( Γ ) ⩽ 1 , and Γ is inconsistent iff consistn(Γ) = 0. Lastly, the concept of consistency degrees of theories is generalized and a method of graded reasoning in L n ∗ is obtained.

It is shown that Cech completeness, ultra completeness and local compactness can be defined by demanding that certain equivalences hold between certain lasses of Baire measures or by demanding that certain lasses of Baire measures have non-empty support. This shows that these three topological properties are measurable, similarly to the classical examples of compact spaces, pseudo-compact spaces and real compact spaces., peer-reviewed

In 1987 V.I. Ponomarev and V.V. Tkachuk characterized strongly complete topological spaces as those spaces which have countable character in their Stone–Čech compactification. On the other hand, in 1998 S. Romaguera introduced the notion of cofinally Čech complete spaces and he showed that a metrizable space admits a cofinally complete metric (otherwise, called ultracomplete metric), a term introduced independently by N.R. Howes in 1971 and A. Császár in 1975, if and only if it is cofinally Čech complete. In a recent paper the authors showed that these two notions are equivalent and in this way answered a question raised by Ponomarev and Tkachuk [Vestnik MGU 5 (1987) 16–19] about giving an internal characterization for strongly complete topological spaces (termed ultracomplete by the authors). In this paper, sums and products of ultracomplete spaces are studied., peer-reviewed

We establish the completeness of spaces of $\mu$-pseudo almost periodic functions (or sequences) and $\mu$-pseudo almost automorphic functions (or sequences) by establishing a new result on the closedness of the sum of closed vector subspaces of the Banach space of bounded functions. To obtain this result we use abstract tools on the closedness of the image of linear operators and the sum of closed vector subspaces of a Banach space.