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*MML* *Identifier*: PENCIL 4.

"... Summary. In this paper we construct several examples of partial linear spaces. First, we define two algebraic structures, namely the spaces of k-pencils and Grassmann spaces for vector spaces over an arbitrary field. Then we introduce the notion of generalized Veronese spaces following the definitio ..."

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Summary. In this paper we construct several examples of partial linear spaces. First, we define two algebraic structures, namely the spaces of k-pencils and Grassmann spaces for vector spaces over an arbitrary field. Then we introduce the notion of generalized Veronese spaces following the definition presented in the paper [8] by Naumowicz and Pra˙zmowski. For all spaces defined, we state the conditions under which they are not degenerated to a single line.

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*MML* *Identifier*: UNIALG_1.

"... The articles [5], [7], [6], [8], [2], [1], [4], and [3] provide the notation and terminology for this paper. For simplicity, we follow the rules: A denotes a set, x, y denote finite sequences of elements of A, h denotes a partial function from A ∗ to A, and n denotes a natural number. Let us conside ..."

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The articles [5], [7], [6], [8], [2], [1], [4], and [3] provide the notation and terminology for this paper. For simplicity, we follow the rules: A denotes a set, x, y denote finite sequences of elements of A, h denotes a partial function from A ∗ to A, and n denotes a natural number. Let us consider A and let I1 be a partial function from A ∗ to A. We say that I1 is homogeneous if and only if: (Def. 1) For all x, y such that x ∈ domI1 and y ∈ domI1 holds lenx = leny. Let us consider A and let I1 be a partial function from A ∗ to A. We say that I1 is quasi total if and only if: (Def. 2) For all x, y such that lenx = leny and x ∈ domI1 holds y ∈ domI1. Let A be a non empty set. Observe that there exists a partial function from A ∗ to A which is homogeneous, quasi total, and non empty. We now state three propositions: (1) h is non empty iff domh � = /0. (2) Let A be a non empty set and a be an element of A. Then {εA} ↦− → a is a homogeneous

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*MML* *Identifier*: WAYBEL13.

"... and [12] provide the notation and terminology for this paper. 1. PRELIMINARIES The scheme LambdaCD deals with a non empty set A, a unary functor F yielding a set, a unary functor G yielding a set, and a unary predicate P, and states that: There exists a function f such that dom f = A and for every e ..."

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and [12] provide the notation and terminology for this paper. 1. PRELIMINARIES The scheme LambdaCD deals with a non empty set A, a unary functor F yielding a set, a unary functor G yielding a set, and a unary predicate P, and states that: There exists a function f such that dom f = A and for every element x of A holds if P [x], then f (x) = F (x) and if not P [x], then f (x) = G(x) for all values of the parameters. One can prove the following propositions: (1) Let L be a non empty reflexive transitive relational structure and x, y be elements of L. If x ≤ y, then compactbelow(x) ⊆ compactbelow(y). (2) For every non empty reflexive relational structure L and for every element x of L holds compactbelow(x) is a subset of CompactSublatt(L). (3) For every relational structure L and for every relational substructure S of L holds every subset of S is a subset of L. (4) For every non empty reflexive transitive relational structure L with l.u.b.’s holds the carrier of L is an ideal of L. (5) Let L1 be a lower-bounded non empty reflexive antisymmetric relational structure and L2 be a non empty reflexive antisymmetric relational structure. Suppose the relational structure of L1 = the relational structure of L2 and L1 is up-complete. Then the carrier of CompactSublatt(L1) = the carrier of CompactSublatt(L2). One can prove the following three propositions:

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*MML* *Identifier*: TOPREAL8.

"... theorem. The main goal was to prove the last theorem being a mutation of the first theorem in [12]. ..."

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theorem. The main goal was to prove the last theorem being a mutation of the first theorem in [12].

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*MML* *Identifier*: MATRIX_4.

"... Summary. This article gives property of calculation of matrices. ..."

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*MML* *Identifier*: MSUALG 4.

"... Summary. This article introduces the construction of a many sorted quotient algebra. A few preliminary notions such as a many sorted relation, a many sorted equivalence relation, a many sorted congruence and the set of all classes of a many sorted relation are also formulated. ..."

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Summary. This article introduces the construction of a many sorted quotient algebra. A few preliminary notions such as a many sorted relation, a many sorted equivalence relation, a many sorted congruence and the set of all classes of a many sorted relation are also formulated.

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*MML* *Identifier*: BVFUNC_9.

"... Summary. In this paper, we have proved some elementary propositional calculus formulae ..."

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Summary. In this paper, we have proved some elementary propositional calculus formulae

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*MML* *Identifier*:E_SIEC.

"... The articles [3], [1], [4], and [2] provide the notation and terminology for this paper. In this paper x, y, X, Y are sets. We introduce G-net structures which are extensions of 1-sorted structure and are systems 〈 a carrier, an entrance, an escape 〉, where the carrier is a set and the entrance and ..."

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The articles [3], [1], [4], and [2] provide the notation and terminology for this paper. In this paper x, y, X, Y are sets. We introduce G-net structures which are extensions of 1-sorted structure and are systems 〈 a carrier, an entrance, an escape 〉, where the carrier is a set and the entrance and the escape are binary relations. Let N be a 1-sorted structure. The functor echaos(N) yields a set and is defined by: (Def. 1) echaos(N) = (the carrier of N) ∪ {the carrier of N}. Let I1 be a G-net structure. We say that I1 is GG if and only if the conditions (Def. 2) are satisfied. (Def. 2)(i) The entrance of I1 ⊆ [:the carrier of I1, the carrier of I1:], (ii) the escape of I1 ⊆ [:the carrier of I1, the carrier of I1:], (iii) (the entrance of I1) · (the entrance of I1) = the entrance of I1, (iv) (the entrance of I1) · (the escape of I1) = the entrance of I1, (v) (the escape of I1) · (the escape of I1) = the escape of I1, and (vi) (the escape of I1) · (the entrance of I1) = the escape of I1. Let us observe that there exists a G-net structure which is GG. A G-net is a GG G-net structure. Let I1 be a G-net structure. We say that I1 is EE if and only if the conditions (Def. 3) are satisfied. (Def. 3)(i) (The entrance of I1) ·((the entrance of I1) \ idthe carrier of I1) = /0, and (ii) (the escape of I1) ·((the escape of I1) \ idthe carrier of I1) = /0. Let us mention that there exists a G-net structure which is EE. Let us note that there exists a G-net structure which is strict, GG, and EE. An E-net is an EE GG G-net structure. In the sequel N denotes an E-net. Next we state several propositions: (1) Let R, S be binary relations. Then 〈X,R,S 〉 is an E-net if and only if the following conditions are satisfied:

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*MML* *Identifier*: WAYBEL19.

"... Let T be a non empty FR-structure. We say that T is lower if and only if: (Def. 1) {(↑x) c: x ranges over elements of T} is a prebasis of T. Let us note that every non empty reflexive topological space-like FR-structure which is trivial is also lower. Let us note that there exists a top-lattice whic ..."

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Let T be a non empty FR-structure. We say that T is lower if and only if: (Def. 1) {(↑x) c: x ranges over elements of T} is a prebasis of T. Let us note that every non empty reflexive topological space-like FR-structure which is trivial is also lower. Let us note that there exists a top-lattice which is lower, trivial, complete, and strict. One can prove the following proposition (1) For every non empty relational structure L1 holds there exists a strict correct topological augmentation of L1 which is lower. Let R be a non empty relational structure. Note that there exists a strict correct topological augmentation of R which is lower. We now state the proposition (2) Let L2, L3 be topological space-like lower non empty FR-structures. Suppose the relational structure of L2 = the relational structure of L3. Then the topology of L2 = the topology of L3. Let R be a non empty relational structure. The functor ω(R) yielding a family of subsets of R is defined by: (Def. 2) For every lower correct topological augmentation T of R holds ω(R) = the topology of T. One can prove the following propositions: (3) Let R1, R2 be non empty relational structures. Suppose the relational structure of R1 = the relational structure of R2. Then ω(R1) = ω(R2). (4) For every lower non empty FR-structure T and for every point x of T holds (↑x) c is open and ↑x is closed.

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*MML* *Identifier*: MULTOP_1.

"... The articles [3], [6], [1], [2], [5], and [4] provide the notation and terminology for this paper. Let f be a function and let a, b, c be sets. The functor f (a, b, c) yields a set and is defined as follows: (Def. 1) f (a, b, c) = f (〈a, b, c〉). ..."

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The articles [3], [6], [1], [2], [5], and [4] provide the notation and terminology for this paper. Let f be a function and let a, b, c be sets. The functor f (a, b, c) yields a set and is defined as follows: (Def. 1) f (a, b, c) = f (〈a, b, c〉).