## Algebraic logic, varieties of algebras, and algebraic varieties (1995)

Citations: | 13 - 5 self |

### BibTeX

@MISC{Plotkin95algebraiclogic,,

author = {B. Plotkin},

title = {Algebraic logic, varieties of algebras, and algebraic varieties},

year = {1995}

}

### OpenURL

### Abstract

Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could be understood as an universal algebraic geometry. This geometry is parallel to universal algebra. In the monograph [51] algebraic logic was used for building up a model of a database. Later on, the structures arising there turned out to be useful for solving several problems from algebra. This is the position which the present paper is written from.

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Citation Context ...identities over arbitrary algebraic structures. Here, algebraic varieties correlated with arbitrary varieties of algebras are considered. We first remind some matters well-known in algebraic geometry =-=[69, 25, 65]-=-. Let P be a field and K – its extension. We consider the ring of polynomials R = P[x1, . . . , xn], and take the affine point space K (n) . If T is a collection of polynomials from R, then it is atta... |

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Citation Context ...ote that the theory we deal with here was stimulated, in considerable extent, by investigations of equations in groups. These investigations, in they turn, are connected with geometrical algebra; see =-=[19, 20, 61]-=-. See also [49] as a survey of works on geometrical algebra, in particular, of the works of E. Rips and Z. Sela. The geometric approach clears some ways for seeking solutions. Generally the aims of al... |

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Citation Context ...ote that the theory we deal with here was stimulated, in considerable extent, by investigations of equations in groups. These investigations, in they turn, are connected with geometrical algebra; see =-=[19, 20, 61]-=-. See also [49] as a survey of works on geometrical algebra, in particular, of the works of E. Rips and Z. Sela. The geometric approach clears some ways for seeking solutions. Generally the aims of al... |

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Citation Context ...e controlled by this algebra. 2. Θ-logic. Θ-logic is built up on the ground of some variety of algebras Θ. Since we have in mind many-sorted algebras as well, let us recall some related concepts (see =-=[40, 28, 3, 5]-=-). First of all, we fix a set of sorts Γ. Correspondingly, we consider a many-sorted set G = (Gi, i ∈ Γ). Each Gi is the domain of the sort i. Denote by Ω a set of operation symbols. Each ω ∈ Ω has a ... |

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Citation Context ...nonempty, but the situation changes for G-groups. Along with generalized equations, generalized identities can be considered. The literature on generalized equations and identities is quite extensive =-=[61, 46, 47, 36, 37, 57]-=-. We still shall make some remarks on the closure of a point. We shall make it apparent that if equations admit solutions in a G-algebra G, then every point µ: G ∗ W → G coincides with its closure. We... |

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Citation Context ...use the word “calculus” for the first-order Θ-logic). There are also algebraic equivalents of nonclassical first-order logics [17]. Similar constructions are developed for other logics [12]. See also =-=[6, 9, 7, 8, 10, 11, 66, 67, 68, 62, 63]-=-. We consider the respective algebraizations of Θ-logic. This generalization is necessary for databases with the data type Θ, and for algebra itself as well. We confine ourselves with Halmos algebras.... |

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Citation Context ...eover, implicative classes are discussed in [58], and the rules of inference include also this one: from u0 → 0, infer u0 → (w ≡ w ′ ) for every w and w ′ of the same sort. On this question, see also =-=[30, 64, 27]-=-. 2. Quasigroups. We shall consider here a known problem in the quasigroup theory. A quasigroup is a group without associativity and, of course, without unit. If the unit is added, then we have a loop... |

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Citation Context ...e controlled by this algebra. 2. Θ-logic. Θ-logic is built up on the ground of some variety of algebras Θ. Since we have in mind many-sorted algebras as well, let us recall some related concepts (see =-=[40, 28, 3, 5]-=-). First of all, we fix a set of sorts Γ. Correspondingly, we consider a many-sorted set G = (Gi, i ∈ Γ). Each Gi is the domain of the sort i. Denote by Ω a set of operation symbols. Each ω ∈ Ω has a ... |

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Citation Context ...eover, implicative classes are discussed in [58], and the rules of inference include also this one: from u0 → 0, infer u0 → (w ≡ w ′ ) for every w and w ′ of the same sort. On this question, see also =-=[30, 64, 27]-=-. 2. Quasigroups. We shall consider here a known problem in the quasigroup theory. A quasigroup is a group without associativity and, of course, without unit. If the unit is added, then we have a loop... |

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Citation Context ...deal with here was stimulated, in considerable extent, by investigations of equations in groups. These investigations, in they turn, are connected with geometrical algebra; see [19, 20, 61]. See also =-=[49]-=- as a survey of works on geometrical algebra, in particular, of the works of E. Rips and Z. Sela. The geometric approach clears some ways for seeking solutions. Generally the aims of algebraic geometr... |

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Citation Context ...rule, we shall use the word “calculus” for the first-order Θ-logic). There are also algebraic equivalents of nonclassical first-order logics [17]. Similar constructions are developed for other logics =-=[12]-=-. See also [6, 9, 7, 8, 10, 11, 66, 67, 68, 62, 63]. We consider the respective algebraizations of Θ-logic. This generalization is necessary for databases with the data type Θ, and for algebra itself ... |

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Citation Context ...in the algebra G. Then we arrive at the universal database (FG, U, VG). If δ: G → G ′ is a surjective homomorphism, then it produces an injective database homomorphism For all f ∈ FG ′ and u ∈ U, See =-=[51, 53]-=-. All this will be in use in §5. δ∗: (FG ′, U, VG ′) → (FG, U, VG). (f ∗ u) δ∗ = f δ∗ ∗ u. §3. Algebraic varieties and varieties of algebras 1. Basic concepts. The present and next section relate to t... |

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Citation Context ...→ (w ≡ w′ ). This is a universal formula of a particular kind. We are interested in the question how to obtain the closure of a set of quasi-identities. The problem was investigated by R. Quackenbush =-=[58]-=-. We translate his result in terms of the algebra U. We rewrite u in the form u0 → (w ≡ w ′ ), where u0 is (w1 ≡ w ′ 1 ) ∧ · · · ∧ (wn ≡ w ′ n). 1.3. Theorem. The set of quasi-identities T is closed i... |

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Citation Context ...identities over arbitrary algebraic structures. Here, algebraic varieties correlated with arbitrary varieties of algebras are considered. We first remind some matters well-known in algebraic geometry =-=[69, 25, 65]-=-. Let P be a field and K – its extension. We consider the ring of polynomials R = P[x1, . . . , xn], and take the affine point space K (n) . If T is a collection of polynomials from R, then it is atta... |

2 |
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Citation Context ...use the word “calculus” for the first-order Θ-logic). There are also algebraic equivalents of nonclassical first-order logics [17]. Similar constructions are developed for other logics [12]. See also =-=[6, 9, 7, 8, 10, 11, 66, 67, 68, 62, 63]-=-. We consider the respective algebraizations of Θ-logic. This generalization is necessary for databases with the data type Θ, and for algebra itself as well. We confine ourselves with Halmos algebras.... |

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Citation Context ...use the word “calculus” for the first-order Θ-logic). There are also algebraic equivalents of nonclassical first-order logics [17]. Similar constructions are developed for other logics [12]. See also =-=[6, 9, 7, 8, 10, 11, 66, 67, 68, 62, 63]-=-. We consider the respective algebraizations of Θ-logic. This generalization is necessary for databases with the data type Θ, and for algebra itself as well. We confine ourselves with Halmos algebras.... |

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Modal Polyadic Algebras
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Citation Context ...se approaches are based on deep analysis of calculus (as a rule, we shall use the word “calculus” for the first-order Θ-logic). There are also algebraic equivalents of nonclassical first-order logics =-=[17]-=-. Similar constructions are developed for other logics [12]. See also [6, 9, 7, 8, 10, 11, 66, 67, 68, 62, 63]. We consider the respective algebraizations of Θ-logic. This generalization is necessary ... |

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Citation Context ...solved some years ago by A.A. Gvaramia during his postdoc in Riga, and even in a more general setting–for an arbitrary axiomatizable class of quasigroups. The solution was given using Halmos algebras =-=[22, 23]-=-. I bring here the sketch of the solution. First, together with the category of quasigroups with homotopies as morphisms, we also take the category of the three-sorted quasigroups. Its objects have th... |

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Citation Context ...solved some years ago by A.A. Gvaramia during his postdoc in Riga, and even in a more general setting–for an arbitrary axiomatizable class of quasigroups. The solution was given using Halmos algebras =-=[22, 23]-=-. I bring here the sketch of the solution. First, together with the category of quasigroups with homotopies as morphisms, we also take the category of the three-sorted quasigroups. Its objects have th... |

1 |
Fully invariant algebraic closure systems of congruences and quasivarietes of algebras
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Citation Context ...eover, implicative classes are discussed in [58], and the rules of inference include also this one: from u0 → 0, infer u0 → (w ≡ w ′ ) for every w and w ′ of the same sort. On this question, see also =-=[30, 64, 27]-=-. 2. Quasigroups. We shall consider here a known problem in the quasigroup theory. A quasigroup is a group without associativity and, of course, without unit. If the unit is added, then we have a loop... |

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