论文标题
含义的族群和伯克霍夫系统
Implication Zroupoids and Birkhoff Systems
论文作者
论文摘要
一个代数$ a = \ langle a，\ to，0 \ rangle $，其中$ \ to $是二进制的，$ 0 $是一个常数，如果满足身份：$（x \ to y to y to y to y to y）\ to z \ of z \ of（y of z \ y x'\ x'\ y x'\ y x'\ y \ y x'\ y \ y \ y \ y \ x' \至0 $，$ 0''\大约0 $。这些代数概括了摩根代数和$ \ lor $ -semilattices，零。让我表示各种含义的Zroupoids。有关导致这些代数的动机的详细信息，我们将读者推荐给[SAN12]（或本文末尾提到的相关论文）。对[SAN12]始于[SAN12]的I子变量晶格的结构进行的研究一直持续[CS16A，CS16B，CS17A，CS17B，CS17B，CS18A，CS18B，CS19]和[GSV19]。本文是该系列论文的续集，致力于对含义的Zroupoid理论做出进一步的贡献。 身份（br）：$ x \ land（x \ lor y）\大约x \ lor（x \ land y）$称为birkhoff的身份。本文的主要目的是证明，如果a是多样性i的代数，则派生的代数$ a_ {mj}：= \ langle a; \ land，\ lor \ rangle $，其中$ a \ land B：=（a \ to b'）'$和$ a \ lor b：=（a'\ land b'）'$，满足Birkhoff的身份。结果，我们表征了Zroupoid A的含义A_ {MJ} $是Birkhoff Systems。这也是从主要结果中得出的，有一些双元素不是双质的，而是满足Birkhoff的身份，这比Birkhoff Systems的“ Birkhoff Bisemigroups”更为笼统，因为Birkhoff Bisemigroups是BiSemogroups的BiSemigroups，它满足了Birkhoff的身份。该论文在伯克霍夫（Birkhoff Bisemigroups）上以一个公开的问题结束。
An algebra $A = \langle A, \to, 0 \rangle$, where $\to$ is binary and $0$ is a constant, is called an implication zroupoid (I-zroupoid, for short) if A satisfies the identities: $(x \to y) \to z \approx ((z' \to x) \to (y \to z)')'$, where $x' := x \to 0$, and $0'' \approx 0$. These algebras generalize De Morgan algebras and $\lor$-semilattices with zero. Let I denote the variety of implication zroupoids. For details on the motivation leading to these algebras, we refer the reader to [San12] (or the relevant papers mentioned at the end of this paper). The investigations into the structure of the lattice of subvarieties of I, begun in [San12], have continued in [CS16a, CS16b, CS17a, CS17b, CS18a, CS18b, CS19] and [GSV19]. The present paper is a sequel to this series of papers and is devoted to making further contributions to the theory of implication zroupoids. The identity (BR): $x \land (x \lor y) \approx x \lor (x \land y)$ is called the Birkhoff's identity. The main purpose of this paper is to prove that if A is an algebra in the variety I, then the derived algebra $A_{mj} := \langle A; \land, \lor \rangle$, where $a \land b := (a \to b')'$ and $a \lor b := (a' \land b')'$, satisfies the Birkhoff's identity. As a consequence, we characterize the implication zroupoids A whose derived algebras $A_{mj}$ are Birkhoff systems. It also follows from the main result that there are bisemigroups that are not bisemilattices but satisfy the Birkhoff's identity, which suggests a more general notion, than Birkhoff systems, of "Birkhoff bisemigroups" as bisemigroups satisfying the Birkhoff's identity. The paper concludes with an open problem on Birkhoff bisemigroups.
